If you check during your turn, then raise when someone behind you has bet, this is known as a check-raise. It is, very rightly, considered to be one of the most aggressive things you can do on a card table (but not as bad as, say, throwing your drink in someone’s face).
You should check-raise on the flop in just a few, very specific circumstances:
You have a very good hand, but the number of players currently playing is too big and needs to be reduced.
You’re playing against someone whom you know you’re beating and you’re certain he’ll call and be the first person to bet next round. If he will not be the first to bet next round, you should simply call and raise on the turn — you get an extra bet out of him that way.
You have the best hand, it’s most likely going to remain the best, and you’re playing at a table full of maniacs who will most likely cap all betting rounds, until … you pull down the largest pot of your life! This is a level of thrill that is virtually unmatched in Poker.
You think a check-raise will very likely work as a bluff now and get your timid competitor to fold before this same little stunt doubles in price next round.
You think you can get a “free card” by check-raising.
You know that someone is playing for a draw, and you’re beating him right this second, but you’ll lose the hand if he’s successful in getting the card he needs. The reason to raise after checking is because you may be able to get him to fold right now — and even if you don’t, you want to make him pay as dearly as possible to see each following card.
In any other situation, you’re better off just calling and waiting for the next round. Check-raising always draws attention from even the sleepiest of players at the table, and it may well not be attention that you want.
The best hand does win a pot, you don’t have to bet it repeatedly to make it better, and on the flop round (which is only half-price), you very likely are losing extra money.
If you think of a bet-raise-and-call being worth 4 units from a total of two people after the final call (bet 1 + call 1 + raise 1 + call 1), that same play is worth 8 on the turn (bet 2 + call 2 + raise 2 + call 2). Even if your opponent folds on a check-raise on the turn, you’ve still made an extra 2 units from what you would have on a folded check-raise on the flop. (You would have called your opponent’s single bet on the flop and then raised his 2-unit bet on the river — he put in 3, rather than 1.)
Check-raising definitely does have its place, but usually you’ll find it later in the game.
No-Limit is a bit different. You want to consider using a very heavy check- raise on any opponent who’s on a draw. The problem here, of course, is that you have to be dead certain that he is drawing (instead of just having you flat- out beaten), and there are many players who will semi-bluff back at you with an all-in.