He is generally referred to simply as "Khan".
Khan was one of the unfortunate consequences of late 20th century eugenics. After the devastating Eugenics Wars of the 1990s, Khan was exiled along with his followers on the space ship S.S. Botany Bay.
In the 2260s, the Botany Bay was discovered adrift by the starship Enterprise under the command of James T. Kirk. Upon reviving from cryogenic freeze, Khan plotted a mutiny to allow him to seize the Enterprise and begin anew his dreams of conquest. Aided by the ship's historian, Lt. Marla McGivers, Khan nearly succeeded in his goal, but was thwarted by Kirk and his crew. Unable to find justice in simply bringing the castaways back for trial, Kirk instead marooned them and McGivers on Ceti Alpha Five, a habitable planet where they could start a new life.
Fifteen years later, the starship Reliant, testing various worlds to serve as proving grounds for the Genesis project, was hijacked by Khan and his surviving crew. The captain and his first officer (former Enterprise crewman Pavel Chekov) were captured by Khan, learning that the world they had beamed to was not Ceti Alpha Six, as they had supposed, but rather Ceti Alpha Five. Ceti Alpha Six had exploded six months after the Botany Bay's crew's arrival on Ceti Alpha Five, throwing the latter planet into the orbit of the former and rendering it nearly uninhabitable. After that many of Khan's crew - including McGivers, who was by then Khan's wife - were killed by Ceti Eels, the only other surviving lifeform. Khan blamed Kirk for marooning him on the planet, and vowed to exact his revenge from his foe.
After seizing control of the Reliant and its crew, Khan attempted to steal the Genesis device, capable of creating a new world from any suitable mass--the ultimate terrorist weapon. He was engaged by Kirk, now an Admiral, and once again in command of the Enterprise. However, Kirk's penchant for playing by his own rules backfired, and Khan nearly destroyed the Enterprise; Kirk escaped only through superior knowledge of how Starfleet vessels worked.
Khan eventually obtained the Genesis device, but was defeated by the Enterprise in space combat in the Mutara Nebula, and was badly injured. As a last resort, Khan attempted to use the Genesis device to destroy both ships, but Mr. Spock was able to repait the Enterprise's warp drive, allowing it to escape (though Spock died as a result of radiation exposure).
Khan died when the Genesis device detonated.