This achievement requires you to win a game of chess using each piece (not including the King).
Queen, Rook, Bishop, Knight and Pawn
Queen and Rook checkmates are simple to do, most games you play will be won using these pieces.
Back Rank mates are common ones.
Using 2 Rooks or 2 Queens or Queen and Rook to pin the enemy king to the side of the board is another easy one.
It is possible to checkmate the enemy king with a King and Queen and a King and Rook if no other pieces are left on the board.
Bishops are diagonal movers only, it's a lot harder to do than using a Queen. It is impossible to get a checkmate on the enemy king with just ONE bishop and a king in the endgame where no other pieces are left on the board.
It can be however if there is any more pieces in addition to the bishop in your army, even a pawn if you are crafty enough though you are better off trying to promote it.
Bishops can be valuable in the end game because it can get around the board quickly along long diagonals making it easy to be involved in a checkmate.
Bishop checkmates are also common in the early stages of a game, because there are more pieces which could wind up blocking the enemy king from escaping whether it be his own pieces or enemy pieces.
Knight checkmates are more complicated but remember Knights can jump pieces and are much more flexible in the early stages of the game.
Because Knights can jump pieces, it is possible to do what is called a "Smothered mate" which means the enemy king is hemmed in by its own pieces, cannot move and is attacked by an enemy knight which cannot be captured.
As the game goes on, a Knight is less useful because it cannot get across the board as quickly as the rest of the pieces could, it could even in some situations be less valuable than a pawn! Knight checkmates are hard to do in the end game as it relies heavily on another piece assisting the checkmate so try and do it earlier in the opening stages or mid stages if you can help it. Just make sure it is the Knight making the attack on the King.
In the endgame, Knights are not easy to get checkmates with.
It is not possible to checkmate the enemy king with ONE Knight and a King if no other pieces remain on the board. (2 Knights and King is possible but it is not forced upon the enemy king).
Probably the hardest one to do is a Pawn checkmate, it must be done while it is a pawn and not a promoted pawn as that is technically a completely different piece.
Pawn checkmates are going to be heavily assisted by other pieces and will probably be similar to a "Smothered mate" as mentionned above, the only difference is the pawn must be directly attacking the King and the King cannot move anywhere else other than capturing the pawn but the pawn is protected by another piece.
It is not possible to checkmate the enemy king with ONE pawn and king (without promoting the pawn) if no other pieces remain on the board.
2 pawns and king is possible provided the 2 pawns are on adjacent columns.