The Balrog that haunted Moria in J.R.R. Tolkien’s legendarium is called Durin’s Bane.
Durin’s Bane was a surviving Balrog that hid beneath the Misty Mountains after the War of Wrath. The dwarves of Khazad-dûm accidentally awakened it while mining for mithril. It killed Durin VI, the king of Khazad-dûm, and forced the dwarves to abandon their ancient city. Afterward, the place became known as Moria.
The Fellowship encounters Durin’s Bane in The Lord of the Rings. Gandalf confronts the creature on the Bridge of Khazad-dûm, where he declares that the Balrog will not pass. Their battle continues from the bridge into the depths and eventually to the mountaintop of Zirakzigil.
Gothmog is a different Balrog name in Tolkien’s mythology, associated with the First Age and the fall of Gondolin. That distinction is a common source of confusion. Durin’s Bane is not given a personal name like Gothmog; its title describes its relationship to Durin and the dwarves.