Speedrunning is the practice of completing a video game, in whole or in part, as fast as possible.
There are many kinds of speedruns (runs, for short). How, for instance, do you define completion of a particular video game? Do you have to defeat the final boss? Do you just have to reach the credits or "The End" screen? Do you collect every collectible item, or only as many as you need to finish? Are you required to play the game in one sitting, or can you take the sum of your best times on each level/section? Do you use the in-game timer, or do you measure real time? Are you allowed to die or take damage if it saves time?
The speedrunning community thus has a lot of terms and distinctions. You can have a Real-Time Attack (RTA), which minimzes real time, or a run that minimzes in-game time: either a single segment run, which requires the game to be beaten in one try, or a segmented run, which allows you to reload previous saves made in the run, to give you multiple attempts. You can aim to simply complete the game as fast as possible, called any%, or you can also achieve 100% item collection, appropriately called 100%. For some games, you can also define other speedrunning "categories" to provide alternative objectives, such as low%—you have to complete the game with the minimum number of items absolutely required to finish—or glitchless/warpless—you cannot use certain game-breaking glitches or in-game warps (usually decided by the community) to finish.
Speedrunning involves a lot of planning, practice, and skill. First, you have to plan your route through the game, usually testing many strategies to see which achieves the overall fastest results. These routes can involve things like performing game events or cutscenes out of the intended order or skipping them entirely, called sequence breaking, or performing and exploiting glitches, minor (or major!) faults or bugs in the game that allow you to perform unintended actions, such as going out of bounds or acquiring items that can otherwise not be obtained easily/at all. Other things to consider include the manipulation of enemy AI patterns; abusing game physics to acquire more movement speed or height than intended; relying on or even manipulating in-game luck to get certain items or successfully perform certain tricks; and reducing/eliminating/exploiting lag, which is slowdown that occurs when there is too much activity occurring in the game for the hardware to handle at normal speed.
There is also a branch of speedrunning called tool-assisted. A tool-assisted speedrun (TAS) will use tools such as savestates (freezing the state of the game momentarily to save a copy that can be later reloaded), slowdown (slowing down the game to arbitrarily slow speeds), and memory watching (checking values of internal game counters not otherwise available to players) in order to create input files that complete the game in as optimized a fashion as possible, often much faster than possible in real time. This is generally considered very separate from normal speedrunning, because effectively all of the difficulty of execution is eliminated, making the entire run an exercise in route planning.
The first game to be popularly speedrun was Doom, with the introduction of online leaderboards tracking user-submitted "demos" of their completing a certain objective in a game. Speedy completion became a popular objective in late 1994, with the introduction of the COMPET-N website. People also began recording speed demos in the game Quake in 1996, and this Quake community eventually culminated in the merger of two popular Quake running homepages into the Speed Demos Archive, a website which is now an expansive archive of peoples' submitted speedruns. Other early speedrunning games were Super Metroid and Metroid Prime, which were very suited to running by virtue of having plenty of opportunity for sequence breaking and a very extensive physics system allowing for many different kinds of movement.
Today, the speedrunning scene is huge, with an enormous number of games represented. Many games or series thereof have large dedicated communities. The list is far too large to begin to effectively enumerate, but I can give a small sample of some of the most popular:
Super Mario (all of them), Legend of Zelda (all of them), Metroid (all of them), Sonic the Hedgehog (all of them), Castlevania (all of them), Donkey Kong (all of them), Kirby (all of them), Final Fantasy (all of them), Megaman (all of them), Resident Evil (all of them), Doom, Quake, Goldeneye 007, Dark Souls, Chrono Trigger, and this is only barely scratching the surface.
Speedrunners are generally very friendly and willing to share their knowledge and discoveries. Even if you know next to nothing about speedrunning, you can get connected with a community and get help getting off the ground, with just about any game you want.
Worth mentioning is the Awesome Games Done Quick events: happening yearly in the winter, with a sister event Summer Games Done Quick in the summer. A large number of top speedrunners are invited to the event site for upwards of a week of speedgaming for charity. The events are streamed live across the world, in multiple languages, and all proceeds go to charity, usually to prevent cancer or other medical efforts around the globe.
Some links to follow for more information: Speedrun.com, Speed Demos Archive, SpeedRunsLive, /r/speedrun on Reddit, Wikipedia's Speedrun article.