Short answers, pulled from the story.
A Balrog is a demonic being in Tolkien's mythology, originating as a Maiar spirit corrupted by the evil Vala Melkor before the creation of the world. They are tall, wreathed in fire and shadow, and armed with fiery whips of many thongs. Only a force of comparable power to Gandalf's could destroy one.
Tolkien's conception changed throughout his career. In early writings, a host of a thousand Balrogs is mentioned in the Quenta Silmarillion, and hundreds appear at the storming of Gondolin. A very late marginal note, never published in Tolkien's lifetime, stated that 'at most seven' had ever existed; the Annals of Aman, written as late as 1958, still described a 'host of Balrogs.'
Durin's Bane was the Balrog of Moria, which hid beneath the Misty Mountains for more than five millennia after Morgoth's defeat. The Dwarves of Khazad-dum disturbed it while mining for mithril, and it drove them from their kingdom. Gandalf the Grey defeated it at the peak of Zirakzigil after pursuing it for eight days through the tunnels of Moria and fighting it for two more days and nights on the mountaintop.
Tolkien's final etymology, published in The Silmarillion, derives Balrog as the Sindarin translation of the Quenya form Valarauko, meaning 'Demon of Might.' In Gnomish, one of Tolkien's earlier invented languages, the name parsed as balc meaning 'cruel' plus graug meaning 'demon.' The concept of a fire demon may have originated in Tolkien's scholarly study of the Old English word Sigelwara in the 1930s.
Gothmog held the titles of Lord of the Balrogs, High Captain of Angband, and Marshal of the Hosts. He slew Fingon, High King of the Noldor, in the Fifth Battle, Nirnaeth Arnoediad, and commanded Morgoth's forces at the Storming of Gondolin. He was killed in single combat by Ecthelion of the Fountain; both fell into the king's fountain in Gondolin and drowned.
Peter Jackson's films The Fellowship of the Ring in 2001 and The Two Towers in 2002 depicted the Balrog as a massive winged creature, based on the design of Tolkien illustrator John Howe. Jackson's team described the wings as expressing the creature's satanic, demonic nature. Earlier artists such as Ted Nasmith had depicted Balrogs without wings.