Check
When the king of a player can be taken by a piece of the opponent. One says that the king is in check. For instance, the white player moves his rook to a position such that it attacks the black king. For example, if the black doesn't do anything about it, the rook could take the black king in the next move. We say that the white rook gives check. It is considered good manners to say check when one checks one's opponent. It is not allowed to make a move such that one's king is in check after the move. If a player accidentally tries to make such a move, he must take the move back and make another move.
Checkmate
When the player is in check and he cannot move or make a move such that after the move, the king is not in check then he has made it. The player that has made it has lost the game, and the player that made at him has won the game.
There are three different possible ways to remove a check, number one, move the king away to a square where he is not in check. Number two, take the piece that gives the check. Number three, in case of a check given by a rook, bishop or queen, move a piece between the checking piece and the king.
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