Check

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A king is in check when the square it occupies is under attack any of the opponent’s chessmen, even if those chessmen cannot move.

A chessman attacks all the squares to which it could move if it were capturing.

If your king is in check, you must get it out of check on your next move. You get out of check by capturing the checking piece, interposing a chessman between your king and the checking chessman, or moving your king to a square that is not under attack by any of your opponent’s chessmen.

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Since pawns capture diagonally, the white pawn attacks the square occupied by the black king. The black king cannot move to e8 because that square is attacked by the white king. The black king can move to f7, g7 or g8, thereby allowing the white pawn to advance and get promoted.

Double check

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Above: Black has the move. Below: The white king in double check.

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Check by the black rook alone could be parried by moving the white bishop to e3, and check by the black bishop alone could be parried by moving the white pawn or white knight to f3, but the white king is in double check, and must move to d2 or d3, allowing the black bishop to capture the white queen. Interposing a piece between the checking chessman and capturing the checking chessman are not options because there are two checking chessmen attacking from two different directions.