In contract bridge bidding, the Stayman convention is a reply of 2♣ to an opening bid of 1NT, asking the opener whether he holds a 4-card major (that is, whether he has four spades or four hearts).
The motivation behind the Stayman convention is that with an 8-card major suit fit between the two hands, it is usually better to play the major suit game than the notrump game. Having a trump suit protects against shortness; at notrump the opponent can win tricks with long suits, while at a trump game, their long suits can be ruffed (trumped). A trump suit also presents the possibility of ruffing cards between the two hands.
The player who bids Stayman must have a 4-card major and at least enough strength to invite game. A no trump or major-suit game generally requires 25 total points (high card points plus long suit points) between the two hands. Therefore, if the partnership is using the standard 15-17 HCP range for the 1NT opening bid, responder should have at least 8 total points to bid Stayman. If responder has a 5-card major, he will normally use a Jacoby transfer instead of Stayman.
If opener has four hearts, he answers 2♥. If he has four spades, he answers 2♠. Otherwise, he gives an artificial answer of 2♦. (What if opener has four in each suit? He can bid either 2♥ or 2♠; if it turns out that responder had the other suit, opener can show it later.) Responder can then place the contract:
With a fit (that is, if responder holds four cards in the same suit) and with game-forcing strength (at least 10 total points), he will bid the major game (4♥ or 4♠).
Without a fit but with game-forcing strength, responder will bid the notrump game (3NT).
With a fit but with only invitational strength (8 or 9 total points), responder will invite game by bidding 3♥ or 3♠. If opener is strong enough, he may raise to game.
Without a fit and with invitational strength, responder invites game by bidding 2NT. If opener is strong enough, he may raise to game.
A reply of 3♣ to an opening 2NT is also Stayman; the valid replies are 3♦, 3♥, and 3♠. In SAYC, 4♣ to an opening 3NT is defined as Stayman, but many partnerships define 4♣ in this situation to be the Gerber convention, since slam is quite likely with such a strong opening hand. Finally, a response of 2♣ to a 1NT overcall is Stayman.
Sources:
Learn to Play Bridge software (http://www.acbl.org/learn/ltpb.html)
Mark London's SAYC guide (http://okbridge.com/sayc.txt)