In chess, tempo refers to the time taken by a move. Each move takes one tempo. Thus a player can be said to "lose a tempo" when taking one more move to achieve something than necessary, or to "gain a tempo" when taking only two moves to do what would have taken three by other methods, for example.
A simple example of losing a tempo may be moving a rook from a1 to a5 and from there to a8; simply moving from a1 to a8 would have achieved the same result with a tempo to spare. Such maneuvers do not always lose a tempo however — the rook on a5 may make some threat which needs to be responded to. In this case, since both players have "lost" a tempo, the net result in terms of time is nil, but the change brought about in the position may favour one player more than the other.
Gaining tempo may be achieved, for example, by developing a piece while delivering check, though here too, if the check can be countered by the development of a piece, the net result may be nil. If the check can be blocked by a useful pawn move which also drives the checking piece away, the check may even lose a tempo.
In general, making moves with gain of tempo is desirable. A player is said to have the initiative if they are able to keep making moves which force their opponent to respond in a particular way or limit their responses. The player with the initiative has greater choice of moves and can to some extent control the direction the game takes, though this advantage is only relative, and may not be worth very much (having a slight initiative when a rook down, for example, may be worthless).
In some endgame situations, a player must actually lose a tempo to make progress. When the two kings stand in opposition, for example, the player to move is often at a disadvantage (zugzwang) and so must triangulate in order to return to the same position but with the opponent to move.
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