A TAS—an initialism for 'tool-assisted speedrun' or 'tool-assisted superplay'—refers to the use of an emulator or some other sort of tool to play a game extraordinarily well, or a playback file created in this manner, or the actual run that the playback file describes.
An emulator is a piece of software which is able to play a game—to emulate it—originally made for some hardware console, using the exact configuration of the game's read-only memory as software input. (Such a game ensconced in a software file is referred to as a ROM for this reason.) Ideally, the emulator is perfect, i.e. it exactly describes the hardware of the system; however, a figure cited by byuu, the creator of the bsnes emulator for the SNES, is that it can take as much as 3 GHz power to perfectly emulate the 21.47 MHz SNES. But regardless of how accurate the emulation is, the fact that the emulator is simply a machine for taking the game state and controller input and display the next game state means that one can add in some tricks to make gameplay easier. For instance, the game states can be saved and later restored, creating the rather unimaginatively named savestates, or the actual speed of emulation can be lowered, giving rise to slowdown. With these two abilities—the power to redo any mistake, and superhuman reflexes—one is a gaming god.
But this is only scratching the surface. Some emulators can also track and search through RAM addresses, allowing one to peer at the important internals of the game: with a suitable RAM map for the game, one can have direct access to information, like certain stats regarding player or enemy positions, or the state of the random number generator. By manipulating these, and using them to one's advantage, one can do things like manipulate random drops or chance events. A rather exceptional example of extreme "luck manipulation" can be found in a tool-assisted playthrough of Monopoly, wherein 4 CPUs are systematically bankrupted in a single turn.
Often, TASes will make heavy use of glitches or other unintended side-effects of certain actions, as these did technically exist in the actual game, so they could have been used in a playthrough. For instance, a common trick is that many NES and SNES games will do silly things when you press opposite directions on the D-pad, e.g. Left+Right, since the construction of the pad didn't allow for that action, and game devs therefore didn't take it into account during programming. Another is that most games can't handle button presses at inhuman speeds, so certain actions will go haywire when fired once every frame or every other frame, allowing that sprite of Mario to ascend underwater at speeds more common to space shuttles, or the lauded Rygar to crab-walk along thin air.
A large TAS community exists at TASVideos (http://tasvideos.org/) where runs are submitted, voted upon, encoded into videos, and archived in a hall of fame. There are huge galleries of up-to-date speedruns of games on almost every major retro console, as well as some recent ones; forums for discussion of speedrunning-related things, both tool-assisted and otherwise; handy lists of tricks and various useful data charts for budding TASers of certain games; discourse on how authors create their runs, like what they did what's in the movie and how exactly it was done; and just a generally cooperative, TAS-driven atmosphere. Some of their hosted runs are multiple hours long, so don't expect to get through it all in one sitting, but there is something nostalgic about seeing games you've played and loved getting smashed to bits with cold, calculated efficiency.