Balrogs were originally Maiar, servants of the Valar, that were corrupted by Melkor Morgoth's power. They hoped to become like Sauron, the most terrible of Morgoth's servants. They did not, however, take a position such as Sauron's. They became spirits of fire, with whips of flame and swords of darkness. They were about man-size, yet greater. They looked like flames, and yet like darkness itself at the same moment. Many believe that they were winged.

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.