What is the touch move rule in chess?

'In serious play, if a player having the move touches one of their pieces as if having the intention of moving it, then the player must move it if it can be legally moved. So long as the hand has not left the piece on a new square, the latter can be placed on any accessible square. If a player touches one of the opponent's pieces then he or she must capture that piece if it can be captured. If none of the touched pieces can be moved or captured there is no penalty, but the rule still applies to the player's own pieces (Schiller 2003:19–20).

When castling, the king must be the first piece touched. If the player touches their rook at the same time as touching the king, the player must castle with that rook if it is legal to do so. If the player completes a two-square king move without touching a rook, the player must move the correct rook accordingly if castling in that direction is legal. If a player starts to castle illegally, another legal king move must be made if possible, including castling with the other rook (Schiller 2003:20).

When a pawn is moved to its eighth rank, once the player takes their hand off the pawn, it can no longer be substituted for a different move of the pawn. However, the move is not complete until the promoted piece is released on that square.

If a player wishes to touch a piece with the intention of adjusting its position on a square, the player must first alert their opponent of their intention by saying J'adoube or "I adjust". Once the game has started, only the player with the move may touch the pieces on the board ' ~ http://en.wikipedia.org/wiki/Rules_of_chess 

A common variant which may be applicable only in casual play is to not consider a move completed until the player has pressed his/her clock. In that case the player may change his mind even after moving a piece to a square and releasing his/her hand from the piece. This variant is not recognized by FIDE nor the USCF.