" If my opponent moves a piece which put his King in check, but he didn't realize it and took his hand off the piece what happens now when I show him his mistake? "
In a blitz game you could claim a win, but I'm not sure this is true for a normal game.
Illegal move
A player who makes an illegal move must retract that move and make a legal move. That move must be made with the same piece if possible, because the touch-move rule applies. If the illegal move was an attempt to castle, the touch-move rule applies to the king but not to the rook. The arbiter should adjust the time on the clock according to the best evidence. If the mistake is only noticed later on, the game should be restarted from the position in which the error occurred (Schiller 2003:24–25).
If blitz chess is being played (in which both players have a small, limited time, e.g. five minutes) the rule varies. A player may correct an illegal move if he or she has not pressed his or her clock. If a player has pressed his or her clock, the opponent may claim a win if he or she hasn't moved. If the opponent moves, the illegal move is accepted and without penalty (Schiller 2003:77).[7]
Of course, on www.chess.com it couldn't happen. The system wouldn't allow you make an illegal move so I guess you are talking about a tournament or other OTB game.
If it's a tournament game, I think many tournaments have their own rules, but a common one is that the player who moved his king into check gets a time penalty. I'm just guessing, but I'm pretty sure.
In a casual game, I guess you just go back a move and forget about it.
7.4 a - If during a game it is found that an illegal move, including failing to meet the requirements of the promotion of a pawn or capturing the opponent’s king, has been completed, the position immediately before the irregularity shall be reinstated. If the position immediately before the irregularity cannot be determined, the game shall continue from the last identifiable position prior to the irregularity. The clocks shall be adjusted according to Article 6.14. Article 4.3 applies to the move replacing the illegal move. The game shall then continue from this reinstated position.
7.4 b - three strikes and you're out.
If my opponent moves a piece which put his King in check, but he didn't realize it and took his hand off the piece what happens now when I show him his mistake?
Also say this exact thing happens but I don't catch the mistake either and move one of my men - does the game just go on?? What happens If I see the mistake after a few moves?
Thanks!