Balrogs are Maiar, or primeval spirits, that Morgoth, the Dark Lord of the First Age of Middle-earth, seduced to his service. They assumed a physical form while in Middle-earth that was powerful and dreadful to behold. The word Balrog means "Demon of Might" in Sindarin; the Quenya form is Valarauko. Balrogs were the servants of Morgoth which the following passages aptly describe:
"For of the Maiar many were drawn to his splendour in the days of his greatness, and remained in that allegiance down into the darkness; and others he corrupted afterwards to his service with lies and treacherous gifts. Dreadful among these spirits were the Valarauko, the scourges of fire that in Middle-earth were called the Balrogs, demons of terror." 
"And in Utumno he gathered his demons about him, those spirits who first adhered to him in the days of his splendour, and became most like him in his corruption: their hearts were of fire, but they were cloaked in darkness, and terror went before them; they had whips of flame. Balrogs they were named in Middle-earth in later days." 
A Balrog possibly was of man-shape, although much larger. It had a mane that could be wreathed in flame and it could surround itself with darkness. The best description of a Balrog is given when Gandalf faces it on the Bridge of Khazad-dum:
"Something was coming up behind them. What it was could not be seen: it was like a great shadow, in the middle of which was a dark form, of man-shape maybe, yet greater; and a power and terror seemed to be in it and go before it...Its streaming mane kindled and blazed behind it. In its right hand was a blade like a stabbing tongue of fire; in its left it held a whip of many thongs....His enemy halted again, facing him, and the shadows about it reached out like two vast wings. It raised the whip, and the thongs whined and cracked. Fire came from its nostrils." 
The Balrog's main weapon was the "Whip of Fire" which resembled a a multi-tailed whip that flamed with fire. This weapon was used to breaks the webs Ungoliant, the giant spider, had bound Morgoth with when she tried to consume the Silmarils that Morgoth witheld from her. This weapon is used repeatedly as their favorite weapon. We also know that the Balrogs of the First Age used other weapons which included black axes and maces. The Balrog which fought Gandalf on the Bridge of Khazad-Dum used both a whip of fire and a flaming sword.
The Balrogs were originally Maiar, angelic beings of lessure stature than the Valar (like Morgoth), but immortal beings which nonetheless possessed considerable power. They were chiefly spirits of fire which meant they could control and create fire about them, but they also could control and possibly use "magic". Notice the confrontation between the Balrog of Moria and Gandalf. He tries to hold the door to Balin's Tomb, but the Balrog uses a "counter spell" to open the door which results in the door breaking:
"...I found myself suddenly faced by something that I have not met before. I could think of nothing to do but to try and put a shutting-spell on the door...Then something came into the chamber- I felt it through the door, and the orcs themselves were afraid and fell silent. It laid hold of the iron ring, and then it perceived me and my spell. What it was I cannot guess, but I have never felt such a challenge. The counter-spell was terrible. It nearly broke me. For an instant the door left my control and began to open! I had to speak a word of Command. That proved too great a strain. The door burst in pieces. Something dark as a cloud was blocking out all the light inside, and I was thrown backwards down the stairs." 
Clearly, the Balrog knew that there was a spell on the door and also knew that another coeval spirit put it there. It also used a counter spell to gain control of the door. Although Balrogs are not depicted as conversant this does not mean they are merely strong and brutish. They are Maiar and would therefore have knowledge that in most likelihood surpassed the Elves and certainly all Mortals. Indeed, Morgoth designated the chief Balrog as the High Captain of Angband who was both cunning and deadly.
Balrogs were immensely strong and powerful. Their mere presence was enough to cause fear and inaction in their enemies. Notice the response of Legolas and Gimli when they first see the Balrog in Moria:
"He drew, but his hand fell, and the arrow slipped to the ground. He gave a cry of dismay and fear...But it was not the trolls that had filled the Elf with terror...Gimli stared with wide eyes. 'Durin's Bane!' he cried, and letting his axe fall he covered his face." 
As can be seen, Balrogs were fiercesome opponents and NEVER to be taken lightly.
No, well maybe. As is so typical with Tolkien there are multiple interpretations. From above, we have the encounter of Gandalf and the Balrog at the door to Balin's tomb: Gandalf speaks a spell of holding and the Balrog counters that spell. Frodo hears Gandalf whispering as he lays the spell on the door, yet we are not told specifically if Gandalf heard the Balrog utter the counter spell.
After all, it was on the other side of a stone door. But given that all other "Magic" of this type is spoken it could be construed that the Balrog had to speak the counter-spell.
On the other hand, Tolkien was quite disgusted with an American movie makers' attempt to do a cartoon version of LotR. He made several comments concerning the depiction of characters and elements; one had to do with the Balrog of Moria:
"The Balrog _never speaks or makes any vocal sound_ at all. Above all he _does not_ laugh or sneer..." 
This could mean that the Balrog never speaks - ever. Or it could mean for this particular encounter he does not speak, for dramatic effect. Take your pick.
But from the same passage we have:
"With a terrible cry the Balrog fell forward, and its shadow plunged down and vanished." 
It appears that Balrogs can vocalize a cry; maybe sounds for spells as well?
This is a point of debate. Some people are firmly entrenched with the idea that Balrogs have wings and can fly. This is mainly from the description in the Fellowship of the Ring:
"It stepped forward slowly on to the bridge, and suddenly it drew itself up to a great height, and its wings were spread from wall to wall..." 
However, this statement is preceded by the description given above  that its darkness was LIKE wings. Tolkien was very particular on the use of words. If the Balrog had wings he would not have used the word "like". This usage denotes that its darkness was tangible and looked like wings; not that it actually had wings. We also have a description of how Balrogs moved from the Silmarillion which tends to support the idea that they were land bound and had to walk/run:
"Then suddenly Morgoth sent forth great rivers of flame that ran down swifter than Balrogs from Thangorodrim, and poured over all the plain...In the front of that fire came Glaurung the golden, father of dragons, in his full might; and in his train were Balrogs, and behind them came the black armies of the Orcs in multitudes such as the Noldor had never seen or imagined." 
It would seem that the Balrogs were _running_ with the lava flow although not as fast. If they flew, then Tolkien would not have used the word "run" as a comparison on how fast the lava flowed. We also know that Glaurung was a Great Worm and not a dragon which could fly. This meant that Glaurung was land bound and that the Balrogs were in his train; the implication is that the Balrogs, too, were land bound.
The Balrogs were the Chief Servants of Morgoth in the First Age of Middle-earth. They served as protectors of Morgoth in the case of Ungoliant, the giant primeval spider, and as his captains in the battles against the Elves:
"Deep in forgotten places that cry was heard. Far beneath the ruined halls of Angband, in vaults to which the Valar in the haste of their assault had not descended, Balrogs lurked still, awaiting ever the return of their Lord; and now swiftly they arose, and passing over Hithlum they came to Lammoth as a tempest of fire. With their whips of flame they smote asunder the webs of Ungoliant, and she quailed, and turned to flight..." 
The Balrogs were also sent by Morgoth to battle the Elves in the Fifth Battle, the Nirnaeth Arnoediad (the Battle of Unnumbered Tears), and the Battle for Gondolin.
This question can only be answered by conjecture as Tolkien left no statement to address this issue.
The Balrogs were originally the servants of Morgoth in the First Age, of whom, Sauron was also his servant. The Balrogs and Sauron were all Maiar and somewhat coeval; it is debatable whether a Balrog would recognize Sauron as Morgoth's successor and _serve_ him willingly.
However, if one considers the chronology of events in the Third Age then a certain "relationship" or understanding becomes evident.
The Balrog was awakened by the Dwarves in the year 1980. The Dwarves then fled Moria in the year 1981. The Tale of Years falls silent regarding Moria until the year 2480, when Sauron starts to populate Moria with his creatures, namely the orcs. From this point on, up to the year of the War of the Ring in 3018 there is a cohabitation of the Balrog and the orcs for 538 years!
Certainly Sauron knew of the existence of the Balrog (especially at the point when his orcs entered Moria), and the Balrog seems to have tolerated the presence of Sauron's orcs. Yet I do not believe that Sauron could have controlled or commanded the Balrog - at this time he lacked the One Ring which held a large part of his native power. Without the One Ring, Sauron would not have had the ability to dominate a will as strong as a Balrog.
As such, the Balrog acted just as he did in the story: in accordance with a "common evil" against "good". Gandalf, like the Balrog, was a Maiar, but aligned with the Valar(good), who were the eternal enemies of Morgoth(evil). Since the Balrogs were the chief servants of Morgoth it naturally follows that the Balrog of Moria would have attacked Gandalf. Not only that, but the Fellowship had entered/invaded Moria, and they were armed. This too would naturally lead to a confrontation.
But it is doubtful that Sauron (without the One Ring) could have commanded the Balrog to say attack Lorien or fight against Minas Tirith. The main powers of a Balrog are not against armies of lesser enemies, or in breaking castle walls, but rather against other powerful opponents like Wizards or Elf-lords.
The Balrog's were responsible for killing several of the high kings of the Elves, much to their sorrow. Their death tally is impressive and includes Feanor (the most famous and powerful of the Noldor), Fingon (the most noble of the Noldor), Ecthelion (Protector of Gondolin),and Glorfindel (Elf-Lord of Gondolin; the First Age Incarnation).
Once awakened, the Balrog of Moria killed Durin VI in Khazad-Dum inthe year 1980 and Nain I in the year 1981 Second Age. This same Balrog gave battle to Gandalf on the Bridge of Khazad-dum. It could be said that this Balrog effectively killed Gandalf, while he himself was slain by Gandalf:
"I through down my enemy, and he fell from the high place and broke the mountain-side where he smote it in his ruin. Then darkness took me, and I strayed out of thought and time, and I wandered far on roads that I will not tell..." 
We also have one of Tolkien's letters which further states that indeed, the encounter with the Balrog resulted in Gandalf's death:
"Thus Gandalf faced and suffered death; and came back or was sent back, as he says, with enhanced power." 
Clearly, the Balrogs were the bane of many important and powerful individuals.
Yes. We do know the name of at least one: Gothmog, Captain of Angband, Lord of Balrogs. It was he who slew Feanor and Fingon. And during the Fall of Gondolin, Gothmog slew Ecthelion but was himself slain:
"Thus it was that he (Feanor) drew far ahead of the van of his host; and seeing this the servants of Morgoth turned to bay, and there issued from Angband Balrogs to aid them. There upon the confines of Dor Daedeloth, the land of Morgoth, Feanor was surrounded, with few friends about him. Long he fought on, and undismayed, though he was wrapped in fire and wounded with many wounds; but at the last he was smitten to the ground by Gothmog, Lord of Balrogs, whom Ecthelion after slew in Gondolin." 
"Gothmog, Lord of Balrogs, high-captain of Angband, was come; and he drove a dark wedge between the Elvenhosts, surrounding King Fingon, and thrusting Turgon and Hurin aside towards the Fen of Serech. Then he turned upon Fingon. That was a grim meeting. At last Fingon stood alone with his guard dead about him; and he fought with Gothmog, until another Balrog came behind and cast a thong of fire about him. Then Gothmog hewed him with his black axe, and a white flame sprang up from the helm of Fingon as it was cloven. Thus fell the High King of the Noldor; and they beat him into the dust with their maces, and his banner, blue and silver, they trod into the mire of his blood." 
After the Last Battle of the First Age there were some Balrogs left that escaped the wrath and destruction of the Valar. The exact number of original Balrogs is not known. Those who survived managed to escape as Tolkien describes:
"The Balrogs were destroyed, save some few that fled and hid themselves in caverns inaccessible at the roots of the earth..." 
We also have the following letter that specifically discusses the Balrog of Moria:
"The Balrog is a survivor from the Silmarillion and the legends of the First Age. So is Shelob. The Balrogs, of whom the whips were the chief weapons, were primeval spirits of destroying fire, chief servants of the primeval Dark Power of the First Age. They were supposed to have been all destroyed in the overthrow of Thangorodrim, his fortress in the North. But it is here found (there is usually a hang-over especially of evil from one age to another) that one had escaped and taken refuge under the mountains of Hithaeglin (the Misty Mountains). It is observable that only the Elf knows what the thing is - and doubtless Gandalf." 
Some people have questioned whether Gandalf knew what this thing was and only the Elf, Legolas, recognized the Balrog. The above letter clearly shows Gandalf DID know what it was; whereas the encounter on the stair was behind a door and Gandalf could not see his opponent.
Gimli, however, only knows it as "Durin's Bane". He does not know that a Balrog is an evil spirit left over from the First Age and a servant of Morgoth.
Tolkien has left open the possibility of more Balrogs still buried deep and waiting to be awakened.
As stated, the Quenya singular form is Valarauko, "Demon of Might".
Also used is Durin's Bane by the Dwarves because Durin VI was killed by a Balrog when the Dwarves awoke it in their mining of mithril deep in Moria. Gandalf names the Balrog as Flame of Udun. Nowhere else is this term used, and we are unsure as to its definition. In the Appendix of the Silmarillion, under the entry "tum" we find the word Utumno which is the Quenyan form for Morgoth's fortress of the First Age. The Sindarin form is Udun. The phrase Gandalf used could simply mean "Servant of Morgoth" in this context since Balrogs are spirits of Fire and Udun was Morgoth's stronghold.
The Tolkien Encyclopedia
The Encyclopedia of Middle-earth
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