The king moves one square in any direction. There is also a special move called castling which moves the king and a rook. The king is the most valuable piece—it is illegal to play any move that puts one's king under attack by an opponent piece.
A rook can move any number of squares along a rank or file. A rook is involved in the king's castling move.
A knight moves to any of the closest squares that are not on the same rank, file, or diagonal. (Thus the move forms an "L"-shape: two squares vertically and one square horizontally, or two squares horizontally and one square vertically.) The knight is the only piece that can leap over other pieces.
A pawn can move forward to the unoccupied square immediately in front of it on the same file, or on its first move it can optionally advance two squares along the same file, provided both squares are unoccupied (diagram dots).
A queen combines the power of a rook and bishop and can move any number of squares along a rank, file, or diagonal.