The Rings of Power
Three Rings for the Elven-kings under the sky,
One Ring to rule them all, One Ring to find them,
The verse in the Lord of the Rings makes explicit reference to twenty Rings of Power: nine for Mortal Men, seven for Dwarf Lords, three for Elven Kings, and one for the Dark Lord. These are the Rings of Power made in the Second Age and concern the events of the Third Age.
It has been said that more rings were made but they are of a lesser nature than the Great Rings of Power. Gandalf suggests as much when he explains the history of the Rings of Power to Frodo:
"In Eregion long ago many Elven-rings were made, magic rings as you call them, and they were, of course, of various kinds: some more potent and some less. The lesser rings were only essays in the craft before it was full-grown, and to the Elven-smiths they were but trifles - yet to my mind dangerous for mortals. But the Great Rings, the Rings of Power, they were perilous." 
Another allusion to more rings is made when Sauron invaded Eregion to reclaim by force all the rings he had sought to rule by the forging and wearing of the One:
"There Sauron took the Nine Rings and the lesser works of the Mirdain; but the Seven and the Three he could not find." 
We are not told how many of these lesser rings were made, nor of the nature of their power. Their use is nowhere recorded in the history of Middle-earth.
It is also possible that Saruman himself created a minor ring as is suggested in the Fellowship of the Ring during the Council of Elrond where Gandalf is recounting his capture in Isengard:
"But I rode up to the foot of Orthanc, and came to the stair of Saruman...He wore a ring on his finger." In the same section Saruman declares "For I am Saruman, the Wise, Saruman the Ring-maker, Saruman of Many Colours." 
We are told in several places that the study of the lore of the Rings of Power was Saruman's provence. It is quite possible and plausible that Saruman created his own ring of power and used this ring to control and dominate his army of orcs, wargs, and half-breeds. After all he was a Maia (of Aule, the smith) and possessed great power.
The Elves of Eregion made all the rings, except for the One which Sauron forged by himself in Mount Doom.
After the defeat of Morgoth in the First Age, some of the remaining Noldorin Elves settled in Eregion and built a city called Ost-in-Edhil around the year 750 in the Second Age close to the west gate of the dwarven kingdom of Moria. About the year 1200, Sauron came among the Elves in a fair form using the name Annatar (Lord of Gifts), but with a dark plan to ensare them. Sauron greatly desired to "persuade the Elves to his service, for he knew that the Firstborn had the greater power."  He taught them secret lore, and with this knowledge their craftsmen (a guild called the Gwaith-i-Mirdain, the People of the Jewel-smiths) created the Rings of Power which included the Seven and the Nine.
But Sauron had a part in the creation of these rings and he guided the Elves in their making. However, the Three Elven Rings were conceived and made by the Elven-smith, Celebrimbor, alone, and Sauron never touched the Three.
The reason is tied to the regret the Elves had for the passage of time. The Elves were immortal and were fated to live as long as Middle-earth lasted. As such, the earth changed with the passage of time, and the Elves saw many things that were fair become destroyed and lost by the hurts of evil. Sauron, as tempter, awoke a desire in the hearts of Elves to heal the hurts of the earth and create a paradise on this side of the sea to compare to Valinor - AND to be its rulers; whereas in Valinor they were only subjects and below the Valar. The Rings of Power were primarily made to slow the passage of time and preserve their creations of beauty. Yet they had other powers as well.
Tolkien provides a revealing insight on to the nature of the Rings and their powers in one of his letters:
"The chief power (of all the rings alike) was the prevention or slowing of decay (i.e. `change' viewed as a regrettable thing), the preservation of what is desired or loved, or its semblance -this is more or less an Elvish motive. But also they enhanced the natural powers of a possessor - thus approaching `magic', a motive easily corruptible into evil, a lust for domination. And finally they had other powers, more directly derived from Sauron...such as rendering invisible the material body, and making things of the invisible world visible." 
The Rings were not made as instruments of war or domination; they could not create lightning bolts or hail storms. Yet, they conferred powers commensurate with that of the user; a Great Ring in the hands of a weak and lesser person could not work effects to the extent of the wise or great. Notice Galadriel's words to Frodo in Lothlorien:
"Did not Gandalf tell you that the rings give power according to the measure of each possessor? Before you could use that power you would need to become far stronger, and to train your will to the domination of others." 
The Elves used the Three Rings to create islands of timeless beauty and guard them against the passage of time and evil. Their use can be seen at work at various points :
But the use of the Elven Rings was possible only after Sauron was defeated in the Second Age and his Ring taken and assumed lost. If Sauron regained the One, then all the works of the Elves and the use of their Rings would be subject to the evil will of Sauron.
As stated, the Seven and Nine Rings were originally made by the Elves and not evil until Sauron forged the One and later took these rings by war. Their initial purpose was to slow the passage of time and preserve beauty, but since Sauron had a part in their making they became accursed and had evil powers. He gave the rings to different races of Middle-earth to enslave and so control them. Sauron gave the Seven to the Dwarves, who proved harder to enslave;
"they ill endure the domination of others, and the thoughts of their hearts are hard to fathom, nor can they be turned to shadows. They used their rings only for the getting of wealth; but wrath and an overmastering greed of gold were kindkled in their hearts..." 
This implies their rings had other powers but were not used probably because this would draw attention to the user and all that he did.
Sauron gave the Nine to Mortal Men who proved easiest to ensnare. It was said that
"those who used the Nine Rings became mighty in their day, kings, sorcerers, and warriors of old. They obtained glory and great wealth...They had, as it seemed, unending life, yet life became unendurable to them. They could walk, if they would, unseen by all eyes in this world beneath the sun, and they could see things in worlds invisible to mortal men..." 
According to Letter #131, the Seven and Nine conferred invisibility to the user as well as unending life. Also, the ability to make the wearer invisible seems to be a controllable power - not always "activated", but rather selectable when the wearer desires to be invisible. However, eventually the user would fade and become a wraith under the control of Sauron, the Dark Lord. The Three Elven Rings did not confer invisibility, because this is a power from Sauron, and Sauron did not have a part in the creation of the Three Elven Rings, nor did he ever touch them.
It was part of Sauron's scheme to ensnare and enslave the users of all the rings of power and so control the Noldor of Middle-earth.
Sauron planned for the domination of all of Middle-earth and he needed/wanted to control the Elves to complete this plan. This was the reason for the forging of the One Ring. Sauron went to Orodruin, Mount Doom, to forge the Ruling Ring, and by putting a large part of his own inherent power into the Ring he created a means by which he could enslave the users of the rings:
" And much of the strength and will of Sauron passed into that One Ring; for the power of the Elven-rings was very great, and that which should govern them must be a thing of surpassing potency; and Sauron forged it in the Mountain of Fire in the Land of Shadow. And while he wore the One Ring he could perceive all the things that were done by means of the lesser rings, and he could see and govern the very thoughts of those that wore them." 
Sauron took quite a risk in placing a major portion of his own power into an item that could be taken from his control. This is exactly what happened at the end of the Second Age when the Last Alliance of Elves and Men defeated Sauron and Isildur cut the One Ring from Sauron's hand. Separated from his power, Sauron was vanquished and seemingly disappeared from Middle-earth. Tolkien's view on the use of power is revealed in another one of his Letters:
"The Ring of Sauron is only one of the various mythical treatments of the placing of one's life, or power, in some external object, which is thus exposed to capture or destruction with disastrous results to oneself. If I were to `philosophize' this myth, or at least the Ring of Sauron, I should say it was a mythical way of representing the truth that potency (or perhaps rather potentiality) if it is to be exercised, and produce results, has to be externalized and so as it were passes, to a greater or less degree, out of one's direct control." 
Yet while Sauron wore the One his powers on earth were actually enhanced. Its powers were primarily of domination and control. He used the power of the Ring to rule whole peoples who worshipped him as a God-king. Through its use he could control the orcs, the trolls, the Haradrim, wargs, and his most feared servants the RingWraiths. Because Sauron could already control the weather and the Fires of Mount Doom, these too would have been further enhanced to truly frightening levels if he had regained the One. With the One, his Eye would have seen every thought or action anywhere in Middle-earth; no secret could have been kept from him and no one could have withstood him. With the One, Sauron would have captured the Three Elven Rings and he would have used them to create Timeless evil in Middle-earth; all under the control of the One.
For a brief time, all the Rings of Powers were presumably worn and used in Eregion between the time Sauron left the Elves after the rings were made (1590 SA) and when Sauron forged the One Ring (1600 SA). But as soon as Sauron put on the One Ring, the Elves took off all their rings for they finally saw through the deceptions of Sauron and realized that their creations would only be corrupted and used for evil:
"But the Elves were not so lightly to be caught. As soon as Sauron set the One Ring upon his finger they were aware of him; and they knew him, and perceived that he would be master of them, and of all that they wrought. Then in anger and fear they took off their rings." 
Some counseled that their rings be destroyed but the Elves could not bring themselves to destroy the Rings - and so they were hid. The Seven and the Nine were separated. Of the Three, Celebrimbor gave one, Nenya, to Galadriel, and the other two, Vilya and Narya, to Gil-galad. In turn, Gil-galad kept Vilya and gave Narya to Cirdan of the Grey Havens to hold. Then, when he marched to war in the Last Alliance, he gave Vilya to Elrond. An alternate to this version, given in Unfinished Tales states that during the first Council of the Second Age, it was deemed wise that Elrond receive the ring Vilya and Gil-galad kept Narya until departing to war in the Last Alliance when he then gave it to Cirdan.
Angered that his plan was revealed and failed, Sauron made war on the Elves to claim the Rings without his help they could not have made. He laid Eregion to waste and sacked the Elvish city of Ost-in-edhil. He then captured and tortured Celebrimbor into revealing where the Rings were hidden:
"There Sauron took the Nine Rings and other lesser works of the Mirdain; but the Seven and the Three he could not find. Then Celebrimbor was put to torment, and Sauron learned from him where the Seven were bestowed. This Celebrimbor revealed, because neither the Seven nor the Nine did he value as he valued the Three: the Seven and the Nine were made with Sauron's aid, whereas the Three were made by Celebrimbor alone, with a different power and purpose." 
Using Celebrimbor's body as a battle standard on a pole, Sauron returned with war and overran all of Eriador in search for the Three Elven Rings. But he never found them and only suspected where they were hid. With the help of the Numenoreans, Sauron was finally driven out of Eriador c. 1701 Second Age and there was peace for a long while.
Sauron's power was later challenged by the Numenoreans and he was taken back as prisoner where through the use of the One Ring he corrupted them and incited them to rebel against the Valar. Then Iluvatar was called on and the world was changed and Númenor drowned in the ocean. Sauron went down as well, but his spirit fled (with the Ring) back to Middle-earth. When he took shape again he saw how the Númenoreans had grown in strength where their new kingdoms sprang up. He resolved to make war and drive them out before they became too strong. Then the Last Alliance was formed between Elves and Men to battle Sauron. There, before the Black Tower, Barad-dûr, Sauron was thrown down and vanquished in the year 3441 Second Age.
But after a respite, Sauron's spirit appeared again in the Third Age and threatened the free people in Middle-earth. Around the year 1000 Third Age the Istari, or Wizards, came to Middle-earth to help in the struggle against Sauron. Last to come was Gandalf and Círdan was moved to turn Narya over to Gandalf as he saw that it would be put to better use:
"Take now this Ring, he said; for thy labours and thy cares will be heavy, but in all it will support thee and defend thee from weariness. For this is the Ring of Fire, and herewith, maybe, thous shalt rekindle hearts to the valuor of old in a world that grows chill." 
So at the time of the story of Lord of the Rings, the Three Elven Rings were in the hands of the Wise; Gandalf, Elrond, and Galadriel, the Nine Sauron held, as well as three of the Seven(the other four were consumed by dragons), and the One was in the hands of Frodo the Hobbit.
Saruman's study of the Rings provides us with useful information as to their appearance. Gandalf recounts the history of the One Ring during the Council of Elrond by recalling the words of Saruman:
"The Nine, the Seven, and the Three", he said, "had each their proper gem. Not so the One. It was round and unadorned, as it were one of the lesser rings; but its maker set marks upon it that the skilled, maybe, could still see and read." 
These letters could be seen when the One Ring was heated, as in Frodo's fire where Gandalf confirms his suspicions by making the inscriptions appear as fiery letters of elven script in the Black Tongue.
We also know the types of gems used for the Three Elven Rings:
We know that Nenya was made of mithril from the description at the Grey Havens:
"On her finger was Nenya, the ring wrought of mithril, that bore a single white stone flickering like a frosty star." 
What the other Elven-rings were made of is not known. Perhaps, since they were the creation of Celebrimbor alone and apart and different in nature than the other rings they were all made of mithril. But from the above description it seems as if Tolkien is singling out Nenya as distinct and different by saying "the" ring made of mithril - as if implying that the other rings were made of some other material like gold.
As for the Seven Dwarven Rings, we know that they were of gold where it was said "that the foundation of each of the Seven Hoards of the Dwarf-kings of old was a golden ring."  However, we do not know what gems were used in the rings.
As for the Nine, there is nothing written that describes what they were made of or what type of gems they had. Yet, it could be assumed that they too, were of gold.
We only know that the Three Elven Rings had their own proper names:
NARYA (Fire); also known as the Ring of Fire or the Red Ring, and by one obscure reference as the Kindler (Letter #301).
NENYA (Water); also known as the Ring of Adamant or the White Ring, also called chief of the Three in Unfinished Tales.
VILYA (Air); also known as the Ring of Sapphire or the Ring of Air, also called mightiest of the Three in Return of the King.
Sauron's ring had no proper name, but was referred to by many names: the One Ring, the Ruling Ring, Sauron's Ring, or the Great Ring of Power.
The dwarven rings and those of the Nazgul are only known collectively as the Seven and the Nine, and if they had each their own proper name they are not known.
This is a hotly debated issue. Some believe that the Nazgul kept their own rings and were enslaved and controlled by Sauron through their rings. This side also believes that the Nazgul derived their power as Ringwraiths by wearing their own rings. The only clear statement for this stand is during the Council of Elrond where Gandalf states that "The Nine the Nazgul keep."  There are faults with this position; 1)why was there no ring left on the ground after the Witch-king was defeated by Eowyn and Merry, whereas Sauron's Ring was left for the taking by Isildur when defeated by Gil-galad and Elendil, and 2)why didn't Frodo, as Ring Bearer, see the rings on the Ringwraiths on Weathertop and at the Ford of Bruinen when he saw their swords, pale faces, their eyes, and their crowns? Compare this to the fact that while in Lothlorien he was able to see Galadriel's ring, Nenya, upon her finger when Sam could not.
Most of the evidence points to the opinion that Sauron himself held the rings. Gandalf explains the history of the Rings to Frodo at Bag End and tells him that:
"So it is now; the Nine he had gathered to himself; the Seven also, or else they are destroyed. The Three are hidden still." 
This statement is clearly in context with rings and not wills as mention of the Seven is made. We know of no wraiths or slaves that wear the remaining Seven. Therefore, gathering the Nine and Seven to himself means just that - Sauron has the Nine and remaining Seven Rings in his physical possession.
Galadriel also confirms this by telling Frodo while in Lothlorien:
"You saw the Eye of him that holds the Seven and the Nine." 
Then we have the following accounts from The Hunt for the Ring in Unfinished Tales:
"At length he(Sauron) resolved that no others would serve him in this case but his mightiest servants, the Ringwraiths, who had no will but his own, being utterly subservient to the ring that had enslaved him, which Sauron held."
"They were by far the most powerful of his servants, and the most suitable for such a mission, since they were entirely enslaved to their Nine Rings, which he now himself held..."
These statements clearly show that it was Sauron who held the rings and so controlled the Nazgul.
The most definitive statement comes from one of his letters describing the situation of Frodo at the Cracks of Doom and is extremely informative concerning the nature of One Ring:
"Sauron sent at once the Ringwraiths. They were naturally fully instructed, and in no way deceived as to the real lordship of the Ring...But the situation was now different to that under Weathertop, where Frodo acted merely in fear and wished only to use (in vain) the Rings's subsidiary power of conferring invisibility. He had grown since then. Would they have been immune from its power if he claimed it as an instrument of command and domination? Not wholly. I do not think they could have attacked him with violence, nor laid hold upon him or taken him captive; they would have obeyed or feigned to obey any minor command of his that did not interfere with their errand - laid upon them by Sauron, who still through their nine rings(which he held) had primary control of their wills..." 
It would appear from the above body of evidence that when Sauron first claimed the Nine he then gave the Nine Rings to Mortal Men who used the power of their rings for dominance and control. In the end, they faded and became Wraiths - at which point Sauron took their rings back to hold and so control his most feared servants.
The most widely held position is that with the destruction of the One Ring all the remaining rings also lost their power (including the Three Elven Rings) and became ineffectual against the passage of time for which they were created. The reasoning behind this is because the One Ring was embodied with the power necessary to bind and control ALL the Rings of Power when the One's power was destroyed so was the power of all the other Rings. Those who held the Three; Gandalf, Elrond, and Galadriel departed over the sea into the West and took the Rings with them. The evidence for this position is found in several places:
"Yet many voices were heard among the Elves foreboding that, if Sauron should come again, then either he would find the Ruling Ring that was lost, or at the best his enemies would discover it and destroy it; but in either chance the powers of the Three must then fail and all things maintained by them must fade, and so the Elves should pass into the twilight and the Dominion of Men begin." 
"But when all these things were done, and the Heir of Isildur had taken up the lordship of Men, and the dominion of the West had passed to him, then it was made plain that the power of the Three Rings also was ended, and to the Firstborn the world grew old and grey." 
"But what would happen, if the Ruling Ring were destroyed, as you counsel? asked Gloin. We know not for certain, answered Elrond sadly. Some hope that the Three Rings, which Sauron has never touched, would then become free, and their rulers might heal the hurts of the world that he has wrought. But maybe when the One has gone, the Three will fail, and many fair things will fade and be forgotten. That is my belief." 
The most melancholy reference is when the lady Galadriel explains to Frodo the fate of the Elves upon the outcome of the quest:
"Do you not see now wherefore your coming is to us as the footstep of Doom? For if you fail, then we are laid bare to the Enemy. Yet if you succeed, then our power is diminished, and Lothlorien will fade, and the tides of Time will sweep it away." 
Because the primary power of the Three Rings is to slow or arrest the passage of time, Galadriel's words clearly mean that with the destruction of the One Ring, they too lose their power.
Yes, Sauron willingly went as prisoner to Numenor and took the One Ring with him. He was most delighted to work his evil in Numenor under the guise of a penitent; and he used the Ring to dominate the mind of Ar-Pharazon and to sway the people of Numenor into the worship of Morgoth. His subtle persuasions and twisted whisperings into the ear of Ar-Pharazon of claiming the immortal land of Valinor as his own resulted in the Rebellion of Numenor. But even Sauron was confounded by the magnitude of response to this evil act: the Valar called upon the Creator, Iluvatar, and he opened the seas and sank Numenor:
"The world was broken, and the land was swallowed up, and the seas rose over it, and Sauron himself went down into the abyss. But his spirit arose and fled back on a dark wind to Middle-earth, seeking a home." 
Some have questioned or objected to the idea that if Sauron was destroyed and reduced to a spirit then how was he able to carry off the Ring back to Mordor. One of his letters best addresses this issue:
"Though reduced to a `spirit of hatred borne on a dark wind', I do not think one need boggle at this spirit carrying off the One Ring, upon which his power of domination minds now largely depended." 
One will have to accept the view of Tolkien on this matter, for there is no other plausible mechanism for Sauron, as a disembodied spirit, to return to Middle-earth with the One Ring.
Tom Bombadil is an enigma in Middle-earth; we do not know what he is and his origin is never revealed. Tolkien did state in one of his letters that Tom had been created long before he thought about writing the Lord of the Rings and he put him in because he wanted an adventure for the hobbits along the way to Rivendell. For all that, Tolkien did use Tom to make a statement about the nature of power and domination. Tom is supposed to represent an existence whose key desire is in understanding ONLY. He is not concerned in using this understanding for any purpose. As such power and domination are completely useless to him and have no meaning or effect upon him:
"He is master in a peculiar way: he has no fear, and no desire of possession or domination at all. He merely knows and understands about such things as concern him in his natural little realm. [He represents] the spirit that desires knowledge of other things, their history and nature...and entirely unconcerned with `doing' anything with the knowledge..." 
And from the Council of Elrond:
"Could we not still send messages to him and obtain his help? asked Erestor. It seems that he has a power even over the Ring. No, I should not put it so, said Gandalf. Say rather that the Ring has no power over him. He is his own master. But he cannot alter the Ring itself, nor break its power over others..." 
The Ring cannot effect Tom Bombadil because he is outside the whole issue of Power and Domination; Tolkien uses Tom as an allegory that even this intense struggle between "good and evil" is only part of the whole picture of existence.
The Tolkien Encyclopedia
The Encyclopedia of Middle-earth
Hypertextual Systems by FMI Publishing, 1995