The games on this site are played using Standard Algebraic Notaion (SAN). SAN gives coordinates to each sqare on the chess board:

 a8 b8 c8 d8 e8 f8 g8 h8 a7 b7 c7 d7 e7 f7 g7 h7 a6 b6 c6 d6 e6 f6 g6 h6 a5 b5 c5 d5 e5 f5 g5 h5 a4 b4 c4 d4 e4 f4 g4 h4 a3 b3 c3 d3 e3 f3 g3 h3 a2 b2 c2 d2 e2 f2 g2 h2 a1 b1 c1 d1 e1 f1 g1 h1

Each piece has its own designation letter:
 King: K Queen: Q Rook: R Bishop: B Knight: N
(note that all designation letters are upper case.)
(pawns to not have a designation letter.)

To move a piece, simply specify the piece's designation letter, and the square you wish to move the piece to. For example: Nc3 moves the knight to the c3 square, Ra3 moves the rook to the a3 square. If two or more of the same type of piece can move to the square in question, you must specify either the row or column of the piece you wish to move. For example, if you wish to move one of your rooks to a3, and either rook could move to a3, you would specify either the row or column to unambiguate the move (i.e. Raa3, or R1a3). In the rare event that three pieces can move to the same square, you would need to specify both the row and column of the piece you want to move (i.e. Nc3e4).

Pawns are a bit different. For a simple pawn move, you just specify the destination square. For example e4 moves the King's pawn to the e4 square. To capture a piece with a pawn, you specify the column the pawn is in, an 'x' and the destination square. I.e. exd5 would have the pawn on e4 capture the piece on d5. The notation is the same for an en-passent capture in which exd5 would signify the pawn on e5 capturing the pawn on d5, but the pawn would still move to d6.
If you're moving a pawn to a square which promotes it to another piece, put an '=' after the destination square followed by the designation letter of the piece you wish to promote to. For example: a8=Q, axb8=R, etc.

To castle, use either 0-0 (kingside) or 0-0-0 (queenside).

Some things you can add to make the notation a little more descriptive are a '+' to indicate check, '++' to indicate checkmate, or an 'x' before the destination square to indicate a capture. The following are examples: Ra8++, Bc4+, Rxa2, Rxh8+, etc.
The 'x' is required for a pawn capture, but is optional for all other pieces. The check and checkmate indicators are always optional. Also note that specifying an invalid capture, check, or checkmate does not invalidate a move. For example, if you specify Rxa8++ the move is valid regardless of wether or not checkmate really occurrs.