First off, an introduction to Balrogs: in the beginning, before the world, there was Eru, the one, and also the Ainur. Now Eru declared to them a great music and they sang it. But one of the Ainur started up a discord, for he sought to make his part in the music greater. Some of the lesser Ainu followed him in discord. At the end of the music it was revealed that what they sang of was to be made real, and Eru gave them a vision of the world they had sang about. Now Melko and his servants went into the world, were kicked out, and came back. Among these were Sauron and the Balrogs.
Now skip ahead years before the sun to the First Age, where we see the Balrogs go into battle. Now, in a vale surrounded by mountains on all sides, was the hidden city of Gondolin. This city remained secret to the enemy for many years, for he had no servants, other than bats, that could fly and the city was protected by the Great Eagles. (And Balrogs couldn’t find it because they didn’t have wings and couldn’t fly). But eventually the city was found out by treachery and the Balrogs and Drakes and Orcs came. One of the lords of the city slew Gothmog, lord of Balrogs by driving his helm-spike into its chest and drowning with him in the fountain. Another lord of the city (Glorfindel, the same guy as in the Third Age) was helping escapees from the wreck of the city, when he was impeded by a Balrog, and they both fell to their deaths. (Glorfindel, however, would return more powerful than before)
Finally, the powerful Ainur came from the west and broke the stronghold of the Shadow. In fleeing before the destruction, the Balrogs went to all corners of the earth. The last one we are told of flies to Moria, and ends up being slain by an Old Man who threw it to its death, and in doing so also died.