A Speed Run is exactly what it says. Going through the game in the fastest time possible. Speed runs can then be separated into two categories, Regular or Realtime, where the game plays at normal speed, and Tool Assisted, which generally use an emulator to do things which are too difficult for a normal person to do with any consistency
Examples[]
Quake and Doom are very popular choices for speed runs. Some games reward you for speed runs, such as Final Fantasy IX, which rewards you with the most powerful sword in game, if you get to the section in the final dungeon in 12 hours, and Metroid, which, without spoiling anything, has a different scene at the end