The object of chess is to "checkmate" your opponent's king. Checkmate is when the king has no legal move, and is in danger of capture by one of the enemy's pieces on the next move.
In the diagram below, black's bishop has just put himself in position to capture White's king on the next move, winning the game. White must escape his opponent's check (it is illegal not to do so). If he cannot, the game is over and White is "checkmated." There are 3 ways to stop the opponent's check.
1. Block the check. In the diagram below, White can block the check by moving the rook on g4 to d4. White is no longer in check after that particular move.
2. Capture the checking piece. White could also capture Black's bishop with the rook on f3.
3. Move away from the check. White may move the king to any safe square (not threatened by an opposing piece). White's options are moving the king to c4, b3, and d2. Notice the White king cannot capture the Black knight because the Black rook would capture the king on the next move

Checkmate occurs when one player's pieces put the other's king in check, and makes it impossible for that player to get out of check. The diagram below illustrates a checkmate position, in which White wins the game. Let's review our 3 ways to get out of check.
1. Block the check. In the diagram below, Black has no piece that can fit in-between the White queen and the Black king
2. Capture the checking piece. There is no Black piece that threatens the White queen except the king. Black's king may not take the White queen, because the bishop guards the square that the queen is on
3. Move away from the check. Black's king has no move to escape the White queen's clutches. The queen guards b5, c5, b3, and c3. The White rook guards a3, a4, and a5. The White bishop guards the queen at c4.