A time attack, also called a speedrun or a speed run (both spaced and unspaced show up on searches on Google video), is an attempt to beat a video game in the absolute shortest time possible. With the advent of emulators, this has evolved from players merely rushing through games to a finely tuned art where technicians attempt to shave tenths of a second off of even the more obscure games. Like just about everything else, it seems to support a small, if vigorous internet subculture, with its own body of slang and acronyms.
The first time attack video I saw was probably in the distant days of 2004, when I saw a video of someone beating Super Mario Brothers 3 in around 11 minutes. I was impressed and amused. Only recently, while browsing on youtube, did I find that this video, and many others like it, were not the output of a single ace player, but rather were finely tuned performance art pieces put together by entire teams. Many, if not most of the time attacks are what are called TAS, or tool assisted speedruns, meaning that the player used an emulator where they could save and reload the game if they made a mistake. Emulators also allow the player to exploit glitches in games that might be virtually (but not totally impossible) for someone playing on the original console. Some people would consider this "cheating", but after spending some time studying the subculture around time attack, this is no more cheating than a photographer using time elapse photography is cheating, or Jackie Chan choreographing an exquisite fight scene is cheating. Time attack, in a way, stands on its own, quite separate from a crass objective like "beat a game". Which is perhaps why games that were never even popular still come under the skillful manipulation of time attackers. Jaws, for example, a rather mediocre game, gains a new meaning when you consider that a group of people went out of their way to prove that it could be beaten in under five minutes.
Which isn't to say that the activity doesn't merge in with the interests of the larger community of gamers. The most popular games to time attack are also some of the most popular games, period. The Mario series has been time attacked probably more than any other games, and some games, such as the first one have probably reached the absolute limits of how quickly they can be beaten. Reductions in times for that game are usually measured in frames. The Legend of Zelda, Megaman, Metroid and other popular arcade, adventure and platform games also have been extensively time attacked. Other genres, both earlier and later have undergone the same treatment, although it is not always as popular of an activity. ChronoTrigger, for example, is still several hours long, something that is not as fun to watch as the fast moving replays of Super Mario Brothers.
I have been advocating watching time attack videos on youtube or video.google.com to my friends for a while. It is a little hard to explain the appeal: watching videos of manipulated games. I often mention it in a semi-apologetic manner, mocking myself for sitting in my sweatpants and t-shirt, watching 20 year old games being manipulated for matters of seconds. However, I think that for many people, an hour spent watching someone beat The Legend of Zelda II would perhaps be an hour that would hold their attention surprisingly well.