Some call it the bastardisation of cricket. Others call it fascinating, entertaining, and thought-provoking. Richie Benaud calls it the format that "kids take their parents to".
Yes folks, it's the newest format of cricket to hit the international stage. Formed as a much faster-paced format of the game in the early 2000s it started gaining popularity in about 2005 when the first international game was played in New Zealand. Its rules are very similar to One-Day Internationals: the batting side is allowed twenty overs with which to compile the biggest score possible, then try to bowl out the opposition before they score a higher total. The field is restricted for the first six overs: no more than two fielders are allowed outside a circle that has a diameter roughly half of that of the oval. In addition, no more than five fielders are allowed on the leg side at any time, and no more than five fielders are allowed outside the circle after the first six overs.
The major difference is that the time limit is more strictly enforced. Teams are allowed 75 minutes per innings (roughly three and a half minutes per over), and for every full over bowled past that limit, the bowling team loses six runs (or, rather, concede six runs to the opposition). If the umpires reasonably believe that the batting side is wasting time, or circumstances beyond players' control are forcing time to be wasted, more time is added to the 75-minute limit. Another major change is that of the "free hit" - after a no ball, the delivery immediately after is a free hit for the batting side, meaning they may only go out run out, or via some of the more obscure ways.
Because of limited constraints, batsmen tend to hit out more: ideally, the scoring rate is 8-10 runs per over, which can very quickly amass a score of 160-200. Compare this with One-Day International cricket (with an optimum of 5-6 runs per over) or Test match cricket (where often 300 runs are scored in 90 overs, at about 3.3 per over). With big hitting also comes big mistakes, and wickets are also more commonplace: whereas in Test match cricket, one could expect to see eight or nine wickets taken in a day, a Twenty20 match could see eight or nine wickets fall in the space of an hour.
This format of the game has had several positive impacts on the game of cricket as a whole. First, its largely entertainment-based structure attracts crowds. Cricket tragics will flock to see big hitting from the batsmen, and some people just turn up to watch Natalie Bassingthwaighte or Atomic Kitten. Even so, some fans prefer to watch the game from home, where they are not disappointed: some players are wired up with microphones and commentators can speak to them directly. This usage of mics is most notably used in charity matches, such as the Gillette Fusion match in 2007 (which consisted of Australian celebrities and former cricket players) which raised money for the Shane Warne Foundation. Second, batsmen have much more confidence facing up to bowlers, which has had a greater impact on (particularly) One-Day Internationals. Since the introduction of the format, three One-Day International teams have broken 400 runs in an innings (notably, Australia became the first team to do so, creating a world record - then was beaten in that very same match by South Africa) when previously, the highest score was 396 by Sri Lanka. Third, the condensed time constraints mean that more games can be played in a day. Games last roughly three hours and, assuming a one-hour break between games, three can easily be played between 11:30a and 10:30p at the same ground on the same day. This is not often the case, and usually only two international games are played on the same day at the same ground (if at all), one usually being a curtain-raiser for another (often a women's cricket game before a men's).
Cricketing purists believe that this format of the game is a crock, and criticism has noted the influence the game has had on younger cricketers, i.e. leading them to believe that cricket is a batsman's game and they should not take bowling seriously. I am a cricketing tragic, but not a purist, and I believe that T20 cricket does more good than harm: it generates interest in the game, for one thing; for another, the tactics used in the game are so different and so much more interesting (how do you get a wicket when all the batting side is doing is having a hack?) There is really only one bad thing I can say about Twenty20: "Why THAT name?"
In short, the format is good value for the game of cricket. Expect to see a further surge in popularity in the next few years.