Speedrunning is when an individual attempts to beat part or all of a video game as quickly as possible. This can include individual levels, specific objectives, or unique limitations as decided by the community or player.
any% requires the runner to beat the game as fast as possible.
100% requires the runner to beat the game while collecting all key items or upgrades and doing everything important. 
low% requires the runner to beat the game while skipping every key item or upgrade possible. If skipping everything is the fastest way to beat the game, there will not be a low% category separate from fastest completion. 
Individual level only requires the runner to beat a specific level in the game as fast as possible.
Main article: Tool-assisted Speedrun
Tool-assisted Speedrun is a kind of speedrun that uses emulators to optimise every last frame. They therefore invariably contain feats that are humanly impossible.
Constructing a TAS is more like animating a movie than playing a videogame, and requires much, much more patience than actual videogaming skill. Both TASs and real-time speedrun require concentration, dedication, planning, and far too much free time, but TASs are not listed on this wiki, and you can think of this as the videogaming equivalent of banning steroids. This wiki is about real play.
The most common way of TASing is using (or most would say abusing) slowdown and savestates. Slowdown is when the game's framerate is slowed down to make it run slower, and savestates are used to save the progress suring a project and you can revert to that point if a mistake is made later.