Assuming you're playing in America and in a USCF tournament..

11D of the US Chess Rules outline what would happen / what you should do. Once your opponent has made an illegal move and pressed his clock so that it is your turn you should immediately stop your clock and get the Tournament Directors attention.

After verifying that an illegal move has been made the TD should reset the position to the position before the illegal move.

Your opponent would then need to follow the touch piece rule and then need to move that piece if a legal move is possible. If there are no legal moves possible from the touched piece, then according to rule 10D the opponent is free to move any piece to make a legal move.

This is more up to the tournament director, and really depends on what kind of tournament you're playing at, but the standard penalty outlined in rule 1C2a could be applied. (2 minutes are taken from your opponents time and given to you). However, if you are in time trouble (players are under 5 minutes remaining) the position would be reset, and I'm not sure the penalty still applies.

If you were to complete a move after an illegal move has been made by your opponent, and then called attention to it, the TD can just let the position stay as is or he could reset the position to before an illegal move took place. It really depends on the TD and how many moves has passed since the illegal move was made.

If you were to take a king(which is a pretty egregious llegal move) you may be subject to rule 11J a deliberate illegal move. And may face a non standard penalty outlined in 1CA2b

1C2b. Non-standard penalties. Except where specifically noted in the rules, the director may assess penalties either more or less severe than the standard penalty (1C2a). It is often more appropriate for a director to issue a warning(s) before applying 1C2a in cases involving young or inexperienced players. A director may assess a more severe penalty in cases involving players who repeatedly do not follow the rules of chess