Such an alternate version is known as a chess variant.
The following table shows some game pieces of unorthodox chess, from fairy chess problems and chess variants (including historical and regional ones), and the six orthodox chess pieces.
The columns "BCPS", "Parlett" and "Betza" contain the notation describing how each piece moves.
The notation systems are explained in this page.
Pawn of piece(s) Tamerlane Chess and Full Tamerlane Chess have Pawns that promotes to a certain piece.