A Balrog ("demon of might") is a corrupted Maia from the Lord of the Rings continuum. It is an ancient, evil fire-demon, and only the most powerful can stand against it (for example, Glorfindel and Gandalf).
Balrogs were originally Maiar, corrupted by the influence of Morgoth and transformed into beings of shadow and fire. They served as Morgoth's most powerful soldiers throughout the First Age. The most powerful Balrog was Gothmog, who lorded over the others and served as High Captain of Angband.
The best-known Balrog in fandom is the one who was awakened by mining Dwarves during the Third Age. This unnamed Balrog, called Durin's Bane, encountered the Fellowship of the Ring in Moria, where it dueled Gandalf the Grey to the death.
In the PPC
There have been Sues who slay Balrogs, and even Sues who are Balrogs (sweet, kind, misjudged Balrogs). Neither of these are at all likely. Balrogs are dangerous. Many Sues have been fed to Durin's Bane, however.
The form of the Balrog is used by the Word World as the mini for the works of J.R.R. Tolkien. Mini-Balrogs are just like their Middle-earth counterparts, only about one-twentieth of the size and somewhat less dangerous; at the least, not guaranteed to try to kill their owners, or agents would not use them as pets. Mini-Balrogs do have wings (see below).
It has long been debated among Tolkien fanatics whether or not Balrogs have wings. This is due mostly to a line in The Fellowship of the Ring that describes wing-like formations of shadow surrounding the Balrog in the Mines of Moria.
The wargaming miniatures company Games Workshop makes a Balrog model that has a pair of wings for its licensed Lord of the Rings game, but this obviously cannot be considered even remotely canon.
Unfortunately, unless Tolkien himself one day rises from the grave and declares one way or another what he meant, the debate is likely to continue until the end of time.