[img=http://2wapworld.com/chess/rll.png]
The Rook, moves any number of unobstructed squares up, down, left or right, but not diagonally.
[img=http://2wapworld.com/chess/bll.png]
The Bishop, moves any number of unobstructed squares diagonally. Each bishop can move on black squares only or white squares only.
[img=http://2wapworld.com/chess/qll.png]
The Queen combines the move potential of the Rook and Bishop, so can move any number of unobstructed squares in any straight line.
[xlimg=http://vignette2.wikia.nocookie.net/chess/images/2/26/Chess_Queen_Movement.png]
[img=http://2wapworld.com/chess/nll.png]
The Knight, always moves in an L-shaped pattern by 3 squares and can jump over other pieces (presumably due to being on horseback). Always lands on a different-coloured square to the one it moved from.
[xlimg=http://i.stack.imgur.com/wCxze.png]
[img=http://2wapworld.com/chess/pll.png]
A pawn, normally moves only one square straight forward, however on its initial move can move 2 squares straight forward. If there is an opposing piece diagonally 1 square in front a pawn can capture it, however it cannot capture a piece directly in front, nor move diagonally if no piece is available to capture.
A special move called en passant is sometimes available for pawns. This is possible if a pawn makes an initial move of 2 squares but could have been captured by an opposing pawn if it had only moved 1 square. The opposing pawn is actually able to ignore the fact its enemy moved 2 squares and capture diagonally as if it had only moved 1 square, albeit only if the opposing pawn acts immediately. This is called en passant (in passing).
[xlimg=https://sepetjian.files.wordpress.com/2011/09/en-passant.jpg]
A pawn that reaches the opposite end of the board (the 8th rank for white, or 1st rank for black) can (and must) be promoted to another piece, usually a Queen (but not a King). This is usually highly advantageous for the promoting player.
[img=http://2wapworld.com/chess/kll.png]
The King, moves 1 square in any direction but cannot move to any square where it can be captured immediately. The object of the game is to place the opposing King under attack in such a way that it cannot escape.
A special move is available for the King and Rook called Castling, this can only be played once per game by each side. If the squares between a King and a Rook are vacant, and there are no opposing pieces threatening those squares, and neither the King nor the relevant Rook have moved yet, and the King is not in check, then the King can move 2 squares sideways towards that Rook, while the Rook moves to the other side of the King (2 squares left for Kingside castling, or 3 squares right for Queenside castling).
[xlimg=http://www.learnchessrules.com/images/castmove.png]