Durin's Bane
Balrogs are a powerful demonic race of creatures found in J.R.R Tolkien's fantasy series The Lord of the Rings, they are noted as being a race that have undergone heavy changes by Tolkien as he began to alter them from a race that was once numerous enough to be described as "hosts" to beings that "at most" numbered seven at a time. Another aspect of the Balrog that changed through time was their size and physical nature, with later versions becoming more elemental in nature - fiery and near indestructible (this was the type chosen for the movie, it would appear).
Morgoth
Balrogs were originally Maiar, similar to that of Sauron and Saruman, but after Morgoth's fall into darkness they followed after him and became his demon servants. The creature is reffered to as the Balrog of Morgoth. Morgoth is ultimately the root of all evil in Middle Earth. The demon is encountered in the Mines of Moria, and seems feared by all goblins and orcs that have taken over the tunnels in the mountain.
The Lord of the Rings' Encounter
The Lord of the Rings contains only a brief encounter with the demon; this particular Balrog had no true name and was refered to as Durin's Bane. Durin's Bane lived in the darkness of Khaz-a-dum, a deep place of the world, being older and more monstrous than an orc or goblin it scares off the goblins surrounding the fellowship then follows them to the bridge in Khaz-a-dum. The fellowship of the ring make it across the bridge, but Gandalf stays, and prevents the beast from crossing. He uses his magic to destroy the bridge, but is brought down by the Durin's Bane. The Balrog is eventually slain by Gandalf on the mountains of Charadras.
As seen in Ralph Bakshi's Lord of the Rings.