Bahamut is the last Eidolon summoned by Rydia? in Final Fantasy IV -Interlude-. It is fought in the Tower of Babil after the first four Eidolons are defeated. Rydia? tries to finish off the party with Bahamut, but the real Rydia appears and summon the Mist Dragon to knock some sense into Bahamut who dismisses himself from battle which leads up to the the final boss.
Bahamut (Arabic بهموت Bahamūt) originated as an enormous whale in ancient pre-Islamic Arabian mythology. Upon Bahamut's back stands a bull with four thousand eyes, ears, noses, mouths, tongues and feet called Kujuta (also spelled "Kuyutha"). Between each of these is a distance of a 500-year journey. On the back of Kujuta is a mountain of ruby. Atop this mountain is an angel who carries six hells, earth, and seven heavens on its shoulders.
Another version of the Arabic story is that Bahamut is indeed a dragon and he stands on a whale called Liwash.
In modern times, the game Dungeons & Dragons is responsible for re-imagining Bahamut as the king of dragons, a benevolent Platinum Dragon; the opposite of the malevolent Tiamat, the five-headed Chromatic Queen of Dragons.