You claim a win as per Laws of Chess Appendix A.4b. BTW, if you take the King it would be regarded as an illegal move (1.2) and you will be the one who will lose the game.
In long time control you'll get 2 minutes extra (your opponent's time remains the same) if your opponent made an illegal move for the first time, if it happens second time the game is lost.
Wow, I actually had a win! I did not know about this rule for rapid games, thanks for pointing that out, cavelorum. I should check though if this rule applies to USCF games too. I still feel good about the way I handled the situation: after all, I was down a piece, and I did not want to claim even the draw on a technicality in the unrated tournament, but I sure will use that in the rated games.
I finally checked USCF rules for blitz, and there are differences with FIDE ruling that cavelorum cited: according to USCF rules, if your opponent makes an illegal move while being in check, you may claim a win by either taking your opponent king (here! that's what my intention was in the actual game) or stopping the clock and claiming the win without making any move. In the long time control, when your opponent makes an illegal move, you get extra 2 min on the clock and there is no limit of how many illegal moves your opponent makes, the game continues, you just get 2 min extra for each illegal move your opponent makes. This was an interesting thing to learn, OTB blitz players be aware!
Re Snookslayer: you are right, it is not mandatory to announce checks or checkmates in OTB games. I do announce them when playing much lower rated opponents (under 1400), just out of courtesy. In this case my opponent was about my level, and surely he should have been able to see the check, but I guess he was so focused on finding a quick way to mate me that he missed it.
I recently had a tournament game (OTB blitz) where I delivered a check to my opponent, without announcing it as usually it is not necessary in the OTB games and considering that my opponent was strong, ~1700. He clerly missed the check and made some attacking move that was, of cause, illegal. I was jokingly considering taking his King and announcing a win, however, I simply pointed out that he made an illegal move, and he took it back and proceeded, finishing me off and getting a win. Since this was an unrated tourney, I did not feel too bad about it, but what should I have done in the real rated blitz tournament? In long time controls, you usually call TD and he would penalize the player who made the illegal move by reducing his remaining time in half(?). If that happened in this game, I could have a chance to win on time since we both had only about 2 min left on the clock. Did anyone had this situation and what was the resolution of it?