Tolkien's most widely read works, The Hobbit and The Lord of the Rings, take place entirely in Middle-earth, and Middle-earth has also become a short-hand to refer to the legendarium and Tolkien's fictional take on the world.
The Balrogs were originally Maiar, of the same order as Sauron and Gandalf, but they became seduced by Morgoth, who corrupted them to his service in the days of his splendour before the creation of Arda. During the First Age, they were among the most feared of Morgoth's forces. When his fortress of Utumno was destroyed by the Valar, they fled and lurked in the pits of Angband.
As Maiar, Balrogs would have had the natural ability to change their shape at will, and to move "unclad in the raiment of the world" meaning invisible and without form. As such, they could change their appearance at will. However, it seems that Melkor, Sauron and their Ainur servants all lost the ability to change shape and became permanently bound to one form. Melkor manifested as the "tyrant of Utumno", gigantic and terrible, and was apparently unable even to heal wounds: his hands and forehead remained burned by the Silmarils and his face and foot never healed after the battle with Fingolfin. Sauron became trapped in the form of a gigantic burning man after the flooding of Númenor, and he even lost his finger when the Ring was cut from his hand.
It is unknown whether Balrogs were similarly bound, or if they could change shape.