All rules from standard chess apply, except as explained below:
White makes one move for his first turn. Black makes a series of two moves for his first turn.
White makes a series of three moves for his next turn, then black makes a series of four moves, and so on. Each series has one more move than the previous series.
All moves must be legal in normal chess. If your king is in check, you must move it out of check on the first move of your series. If you cannot, then you are in checkmate and you lose. Likewise you may not move your king into check during your series, even if it moves out of check before the series is over.
If you check the opponent's king, your series ends immediately, even if you have additional moves pending. Your opponent still get all the moves he would normally get. That is, if you check on the third move of turn seven, your turn ends, and your opponent still gets eight moves during his next series.
Enpassant captures are only allowed on the first move of a series. A pawn may be captured enpassant if it made a two-space move on the previous series, and did not move any further.
If a player has no legal moves left in the middle of his series, it is stalemate and the game is a draw.
Additional information on ProgressiveChess can be found at:
Hans Bodlaender's Chess Variant Pages
World Internet Progressive Chess Championship 98 page
Timo Honkela'a Progressive Chess page