The Balrog that confronts Gandalf in Moria is known as Durin’s Bane.
Durin’s Bane was a servant of Morgoth, the first Dark Lord, during the First Age. It survived the destruction of Morgoth’s realm and hid beneath the Misty Mountains. The creature was awakened by the mining of mithril in Khazad-dûm, later called Moria, during the Third Age.
The Balrog killed Durin VI, the dwarf king, in 1980 of the Third Age, which gave rise to its name. It remained in the abandoned underground city until the Fellowship encountered it. Gandalf fought the creature on the Bridge of Khazad-dûm and defeated it after a long struggle atop Zirakzigil, but the battle also caused Gandalf’s death and transformation into Gandalf the White.