'Checkmate' is what you say in chess when you attack the opponent’s king in such a way that no escape is possible, meaning you win the game. It is now used in other contexts to mean 'complete victory'.
The rebels surrounded the president's palace. He had no option but to agree to their demands: it was checkmate.
Unless we make some serious changes around this business our competition will catch up and it will be checkmate.
It can also be used as a verb:
Garry Kasparov checkmated the chess computer after a fine series of moves.
'Stalemate' describes the opposite situation to 'checkmate'. Also coming from chess, a stalemate is when no further action is possible.
I could only afford £150,000 for the house but the seller would go no lower than £155,000. We reached a stalemate and called the deal off.
The pattern of black and white squares on a chessboard is described as being 'chequered' in British English and 'checkered' in American English. The squares themselves are known as 'cheques' or 'checks' respectively. But the verb 'to check' is spelled the same in both places.