After the First Age the Balrogs primarily faded away, and a few lived deep, near the center of the earth. The Dwarves of Moria under Dúrin's rule accidently awoke a sleeping one while digging, and it became known as Dúrin's Bane. In 2480 Sauron sent Orcs and Trolls to Moria, and the Balrog ruled them. Dúrin's Bane was killed by Gandalf after a ten day battle.
The Battle Between Glorfindel and a Balrog:
This, in my opinion, is the most accurate portrayal of what a Balrog looked like.
Paiting by John Howe.