If the move involved his pawns, he likely played a special pawn move called "en passant" which allows a pawn to take another pawn even if it is not directly attacking it (see http://en.wikipedia.org/wiki/En_passant); it is not a bug or cheating.
If the move involved his rooks, he played a move called "castling" (which is shown by 0-0 or 0-0-0 in the notation, and involves the King moving 2 squares toward the Rook, and the Rook moving to the other side of the king - all in one move). See http://en.wikipedia.org/wiki/Castling for more information.
If a pawn makes it all the way to the opposite side of the board, the player can choose to create a knight, bishop, rook or queen. This is called promotion. See http://en.wikipedia.org/wiki/Pawn_promotion for more information.
Sometimes you cannot move what you want to because your king is in check - make sure that nothing is attacking your king! See http://en.wikipedia.org/wiki/Discovered_attack for more information.
If none of these explain the illegal move, you may have found a bug. Please send us the game:
1. Click on your avatar, then "My Games";
2. Navigate to the game with the illegal move;
3. Click on the Send Game to a Friend button;
4. Add [email protected] as the email address to send to;
5. Describe what happened, including move number in the message section;
6. Click on Send.
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