A Brief Guide to the Rules of Chess

The chessboard is placed so that the white square is on the right side of the player.
The player with the White pieces always moves first.
A pawn on reaching the last rank can be exchanged for a Queen, Rook, Bishop or Knight as part of the same move. The effect of this promoted piece is immediate. So if the pawn is promoted to a Queen, the Queen, if it is in a position to do so, may check or checkmate the enemy King.
Each move must be made with only one hand.
A piece that is touched must be moved unless moving this piece would place the King in check. This is called the "touch-move rule".
If an opponent's piece is touched then it must be captured if possible. If this is not possible then play continues as if that piece had not been touched.
A person may adjust a piece on the chessboard on her move by saying "j'adoube" which is French for "I adjust".
When castling, the King must be moved first and then the Rook brought to stand on the opposite side to the King.
When using a clock, the button must be pressed with the hand which moved the piece on the chessboard.
All play should be conducted with respect for the opponent. A player should not distract or annoy her opponent in any way.

For a more comprehensive guide to the official rules of chess see
FIDE Laws of Chess 1997