This is a TAS of the Japanese Super Mario Bros. 2, a.k.a. 'The Lost Levels', using Mario's brother Luigi. He jumps higher than Mario, but acts slower in many ways. However, the movie is in no way worse than the Mario version - some new creative solutions have been found.
This movie uses warps. For a warpless version, see  SNES Super Mario All-Stars: The Lost Levels "warpless, Luigi" by DaSmileKat in 34:45.63.
Watch this run being played back on a real console.