Pieces have the same movement as in orthodox chess, but they cannot take other pieces in the usual way. Instead of the normal captures, pieces can pull or push other pieces, potentially off the board. The goal is to send the enemy king off the board.
Each turn, a player has the following options:
It seems easier to understand with some examples. For a detailed introduction please visit this page (in French).
The e2 pawn can move to e3 and e4 as usual. It can also slide diagonally, being pushed by the bishop or the queen (which may or may not move along this line afterward). It can also go to c3, being pushed by the knight from g1; then the knight can move to e2, or stay motionless. Finally, the pawn can "take the king": this is a special move indicating that you want it to exit the board. Indeed it could be pushed off the board by the bishop or the queen.
Note: if an action is possible but you don't want to play a second part in a move, click on any empty square: this will send an empty move.
It is forbidden to undo a "move + action". For example here, white could push back the black bishop on g7 but not return to d4 then.
Castling is possible as long as the king and rook have not moved and haven't been pushed or pulled (this differs from the chessvariants description).
The game ends when a push or pull action threatens to send the king off the board, and he has no way to escape it.
The king cannot "take" on g4: this would just push the queen one step to the left, and she would then push the king beyond the 'h' file. There are no en-passant captures.
Dynamo chess on chessvariants.com. The short description given on this page might help too.