Turn check raise, an indicator of extreme strength
You have probably experienced that sick feeling in your stomach when you bet the turn in position and get check raised. That feeling should tell you something but the odds are that you have simply ignored it at least a few times. There are few moves in poker that have more power than the check raise, particularly on the turn. As you will see, the reasons are quite obvious, it is simply a matter of not having considered them previously.
If you are check raised on the flop there is still a good chance the other player is bluffing, or that you could still take it away on the turn. When you are check raised on the turn the odds of the other player bluffing are decreased and there is one less opportunity for you to take it away. The money has begun to pile up once you get to the turn and each and every move becomes more and more important. Once you decide to call a check raise on the turn you have all but committed yourself to the hand. It would not be smart to call a check raise with a draw, mainly because you should not allow yourself to be put into a situation where you must decide what to do with a flush draw. If you had a drawing hand and bet the turn after the villain checks you should be willing to get it all in after he check raises you. If you are not comfortable getting your money in with a draw in that spot you would be much better off simply checking back on the turn and seeing the river for free.
This still does not satisfy the question of why the turn check raise is so strong. Think about what hands players are likely to check raise with. Most often they have an extremely strong hand, this is why the move itself is so strong. It gives off a strong vibe to the other player. Either the villain can back down (which would be why a player would bluff check raise) or he will commit his stack (which would be why a player check raises for value.) People tend to get jumpy when they are check raised and as a result they call without considering the consequences of their actions.
Adjusting vs. a villain who you feel is bluffing vs. a villain you feel is check raising for value
Adjusting to a turn check raise is simple. Once you have decided that your opponent is either bluffing or has a strong hand you have a few options. If you determine that he is likely bluffing you call and let him make another bluff on the river. There is no sense in re-raising his check raise if he has a bluff. By doing this you are only scaring away inferior hands.
If you feel that the villain has a strong hand when he check raises you there is more to consider. First, is his hand stronger than mine? If his hand is weaker, but still strong, should I shove all in or just flat call? If his hand is stronger than yours and you do not have the odds to call and see a river that would improve your hand, just fold. If his hand is strong but you likely have it beat, you can either call or re-raise. If the board is draw heavy you are better off getting all in as soon as possible. If there are draws on the board and you flat call, you will allow him the opportunity to hit the draw. In addition, your hand will lose value on the river if he fears that you might have hit your own draw. If the board is not draw heavy you can simply call and allow the other player to force the issue on the river. A re-raise is not a bad play, but if he is going to go all in on the river, why risk him folding on the turn?