Bahamut is a mythical beast whose origin hails from Arabic mythology. He is the vast fish creature that supports the entire Earth on his back and is supposedly so large it can't be seen by human eyes.
In video games, Bahamut has frequently taken on the form of a dragon (or airship) and is a very powerful beast. He is usually characterized by his powerful Mega Flare attack in the Final Fantasy series.
Final Fantasy Series
Bahamut appeared in the Dragon Caves. If you brought him the Rats Tail, you could advance your character classes.
Bahamut appears quite early on in the game when your heroes have just started the legend. Later in the game, when you have received the invincible, you can fight Bahamut as an optional boss.
Bahamut was the Lord of the Summoned Beasts and inhabited the moon, and would allow you to summon him if you bested him in battle. In earlier versions of FFIV you could use Reflect to bounce his MegaFlare back at him. In the DS version he'd just reflect it off himself if you tried that.
Bahamut would appear after getting the first of four stone tablets needed to unlock the legendary weapons. He would then wait for you at North Mountain. As before, you have to beat him to be able to summon him.
Bahamut can be obtained by beating Doomgaze when you get the airship in the World of Ruin. Just don't use the Vanish/X-Zone trick.
There are three versions of Bahamut in Final Fantasy VII, all of which are summoned via materia. Regular Bahamut can be found in the Temple of the Ancients after defeating the area's boss, the Red Dragon. The Neo Bahamut materia is resting on the ground in the Whirlwind Maze, the snowy stretch of land you have to go through to reach the Northern Crater on Disc 2. Neo Bahamut uses an attack called Giga Flare, a more powerful version of standard Bahamut's Mega Flare. The final and most powerful form of Bahamut is Bahamut ZERO. It is obtained by touching the Blue Huge Materia in the Cosmo Canyon planetarium. The player must already have Bahamut and Neo Bahamut in order to receive the summon. Bahamut ZERO's attack is called Tera Flare.
Bahamut can be found in the Deep Sea Research Center, which is located in the southwest corner of the map and can only be accessed via the game's airship, the Ragnarok. Answer his questions correctly and he will decide to test your strength in battle. If you are victorious he joins your party as a GF.
Bahamut is one of Garnet's original Eidolons, Final Fantasy IX's name for summons. However, she is unable to use him or any of her other Eidolons before they are extracted from her on Disc 2. Garnet regains the ability to summon Eidolons by using certain accessories, usually gem stones, in battle. After gaining access to the Hilda Garde III on Disc 3, Beatrix gives Garnet a garnet gem, an item that will finally enable to her summon Bahamut.
In Final Fantasy X Bahamut is the final Aeon you acquire as part of the plot. The player must make it through the labyrinthine Bevelle Temple to obtain him. His special attack is Impulse, which deals moderate non-elemental damage, and his Overdrive, unsurprisingly called Mega Flare, is one of the strongest attacks in the game. Your party, not being the first to pass the trials of the Bevelle Temple, will also encounter hostile Bahamuts controlled by other Summoners over the course of the game.
In the Crystal Bearers, Bahamut appears as the first boss of the game holding a gigantic lance with its tail. Layle eventually kills him by firing swords into the creature, and after "grappling" onto these swords stuck in Bahamut's flesh, grabs the spear with his power and propels it through the beast.
In FFXIII Bahamut is the summon of Fang. In this verison his normal mode looks more humanoid then previous versions. He stands on two legs and has huge pointed claws on his hands. His Gestalt form looks more like a traditional dragon albeit slightly more mechanical.
Bahamut appears as an event monster in the mobile game Terra Battle.