In Tolkien's The Fellowship of the Ring, the type of demon that Gandalf battled in the Mines of Moria. As morgandorf notes below, the balrogs were Maiar corrupted by Melkor/Morgoth. The passage cited seems to indicate that not all Melkor's Maiar flunkies became balrogs, only those whose nature was imbued with fire (whether by Melkor or Eru is not clear). The word "balrog" is Sindarin, rooted in Quenya "valarauko" or "malarauko" ("demon of power"). Compare Latin "val-", as in "valence" and "valor".

In fantasy RPG, drawing heavily on Tolkien for many of the creatures, I have heard of balrog interpretations for at least 5 systems: Dungeons and Dragons, RuneQuest, MERP, Palladium Fantasy, and a Middle-Earth scenario for Sid Meier's Civilization II. In D&D "balrog" is usually synonymous with "Type VI demon".

See also Balrog Award.