Go to Start a new game. Fill in the name and email address of the White player, and the name and email address of the Black player. Choose a password; that's the password that allows you to move. If you don't know the password, you can watch a game, but not move the pieces.
Choose the game you want to see. If you just want to watch, you won't need the password for that game, but if you want to move pieces, make sure you give the game password.
There's a Chess Message Board for discussions about games.
You make moves using standard algebraic chess notation. The columns (files) are labeled A through H and the rows (ranks) are numbered 1 through 8 (from White's perspective).
You make a move by naming the piece you want to move followed by the name of the square you want to move it to. The pieces are named as follows:
Moving the King to d2: Kd2
Moving the Knight to h4: Nh4
If you are capturing a piece, replace the hyphen (-) with an x.
Capturing the piece on d2 with the Rook: Rxd2
If two of the same type of piece (e.g. both Rooks) can make the same move, specify which rook you want to move by naming its file. For example:
Both rooks can legally move to c4. To move the Rook on the A-file: Rac4
Move pawns by simply naming the target square.
Move the E-pawn to e4: e4
Capture with pawns by naming the starting file and the target square.
E-pawn captures the d6 square: exd6
You can castle King-side or Queen-side:
King-side (short) castle: O-O
Queen-side (long) castle: O-O-O
To resign, simply type "resign" or "resigns".
It can't detect checkmate; if you're checkmated, you have to resign.
Finally, to promote a pawn, add a / followed by the name of the new piece to the normal move notation.
Promote a pawn moving to e8 to a queen: e8/Q
Promote a pawn capturing h8 to a Knight: gxh8/N
Thanks to John Watson for the chess program.