From BR Bullpen
A triple play occurs when all three outs in the inning are recorded on the same play. Triple plays are extremely rare occurrences in Major League Baseball. For example, in the 2000s, there were only 35 triple plays in MLB: 5 in 2000, 2 in 2001, 6 in 2002, 2 in 2003, 3 in 2004, 1 in 2005, 5 in 2006, 4 in 2007, 2 in 2008 and 5 in 2009.
Contrary to most double plays, which usually occur on ground balls, triple plays are most often the results of line drives that are caught while runners have left their base, either as a result of a hit-and-run play, a double steal or a spectacular catch on a ball that runners assumed would fall to the ground. In fact, many triple plays are the result of base-running mistakes.
A triple play can also be the result of a ground ball: for example, with no one out and runners on first and second, the batter hits a ground ball to the third baseman playing near the bag; he steps on the base for the first out, throws to the second baseman for the second out, and the second baseman in turns relays to first base for the third out. It sounds simple enough, but it is extremely rare in reality, because the third baseman rarely plays close enough to the bag in such a situation to touch it and throw to second in one motion; if the ball is hit sharply enough, he will throw to second base to start a sure-fire double play rather than try heroically to cut off the lead runner.
Even more rare is the unassisted triple play, where a single defensive player records all three outs. There have only been 15 such triple plays in the regular season in MLB history, one of which took place in the World Series (by second baseman Bill Wambsganss of the Cleveland Indians in Game 5 of the 1920 World Series on October 10, 1920 (Boxscore)). The last unassisted triple play in the Major Leagues was pulled off by second baseman Eric Bruntlett of the 2009 Philadelphia Phillies on August 23 against the New York Mets.
Rarest of all is the unassisted triple play by an outfielder, performed only once in professional baseball history, by Walter Carlisle.
The Minnesota Twins are the only team to have turned two triple plays in one game. This occured on July 17, 1990 (Boxscore) against the Boston Red Sox in a 1-0 win by the Red Sox. The triple plays occured in the 4th and 8th innings.
A 2000 article from Baseball Digest indicates that in 1978, a triple play occurred without contact between bat and ball: the batter struck out, and then the runners on both second base and third base were caught off their bases. A similar triple play occurred on September 3rd, 2002.