In chess, a tempo is a "turn" or single move). It is a way to identify each turn a player spends to move a piece. When a player achieves a desired result in one fewer move, the player is said to "gain a tempo"; conversely, when a player takes one more move than necessary, the player is said to "lose a tempo"). The concept of tempo is important in chess because it is a measure of time and efficiency. Every game simulates a battle between two armies, and as a general of one of those armies, you are in charge of deploying your troops in the most effective way possible. Usually, doing so involves mobilizing your forces without wasting any time.
A move that comes "with a tempo on a piece" is a move that gains a tempo by attacking that piece. For example, in the Scandinavian Defense opening, after 1.e4 d5 2.exd5 Qxd5, Whites 3.Nc3 comes "with a tempo on the queen": the knight attacks Blacks queen, forcing it to move again, and White gains a tempo). Conversely, when one player gains a tempo, it means that the other one has lost a tempo.
In some endgame situations, a player must actually lose a tempo to make progress. For example, when the two kings stand in opposition (a form of zugzwang), the player to move is often at a disadvantage because he must move. The player to move may be able to triangulate in order to lose a tempo and return to the same position but with the opponent to move (and put him in zugzwang). Kings, queens, bishops, and rooks can lose a tempo; a knight cannot).