Please clarify what a continuation bet is in poker, what its purpose is. And if possible, when is it most useful within Texas Holdem?
By definition, a continuation bet is a bet made on one betting round after you were the aggressor in the previous betting round. Continuation bets are typically made regardless of whether or not the cards that came improved your hand. Most often the term applies to bets made during the second round of betting (the flop in Texas Hold'em) after aggression during the first round of betting (preflop in Texas Hold'em).
There are a lot of specific situations that can vary things a lot, but as a general rule, it is best to make a continuation bet when it is likely that your opponent will believe your hand is still better than theirs. This can be a result of the board seeming unlikely to have improved your opponent's hand, a result of the board seeming likely to have improved your own hand, or in some cases a result of your hand being so strong that it did not need improvement.
The key concept to keep in mind about deciding whether or not to make a continuation bet is that you want the story your bets tell about your hand to be believable, particularly if your continuation bet is a bluff.
First, to answer the specific question "What is a continuation bet?"
If you are the aggressor preflop and then are the first to bet on the flop, the bet you made on the flop is a "Continuation Bet." It is called this because had the initiative preflop and are now continuing your aggression postflop. Continuation bets aren't only made on the flop. If you raised the flop and then bet the turn, you've made a continuation bet on the turn. That is all that a continuation bet is -- the questions of what its purpose is and when to do it are much harder to answer.
In order to answer that question why you do it, I think it's best if we first start by thinking about why we make bets in poker to begin with. There are really only 3 reasons to bet in poker:
Almost every bet in poker has either a Value or Bluff component to it, or both. Since you can never have perfect knowledge about your opponents' holdings unless they show their hands down, many times when you bet for Value you also want a few folds as well, and vice versa.
There is a special circumstance which commonly arises in poker that might seem like a reason to bet, but really is just a combination of #1 and #2. That is to force your opponent to surrender his equity in the pot, especially when neither of you have anything. This is one of the most common reasons to make a continuation bet. Consider:
You have KTo on the Button in a No-Limit Holdem game where everybody has 100 BB stacks. One player opens for a limp, and you raise on the button to 5x the BB. The blinds fold, and the limper calls. The flop come A72, rainbow. Your opponent checks to you, and you are confident that he has a very weak hand. What should you do?
Assuming your opponent doesn't have an Ace, you both have garbage hands. Even though both of you have garbage hands, you still both have a certain ammount of equity in the pot. How much depends on your opponent's actual holding. For the sake of argument, let's suppose he has 6h5h. You have the best hand for now, but he has about 6 outs, not counting any redraws. According to the rule of 2 & 4, he has about 25% Equity in this pot, leaving you with 75% equity. If you bet and he folds, he surrenders that Equity to you. If the pot is $100 right now, he is essentially giving you $25 by folding.
On the other hand, what if your opponent has KJo? In this case, it is we who are behind, and now we only have about 3 outs (the Ten), leaving us with only about 15% Equity. Nevertheless, if we bet he is going to fold fairly often and surrender the 85% equity he has to us.
Back to when he has 6h5h. Even if he calls your continuation bet, it's not so bad -- in this case, you got value for your garbage hand because your garbage was better than his garbage; you have the best hand. In the case where he has KJ, we really do want him to fold because he's beating us, so there's an element of bluff to this bet.
From this we can see that given the fact that we never have perfect information our opponents' holdings, every bet we make in poker has some Value and/or some Bluff component in it. Especially in cases where the opponent's range is heavily weighted toward hands that have no absolute value, we also gain by forcing them to surrender their equity. Those cases are classic examples of when a continuation bet should be made.
I'd make a continuation bet in a situation where my hand MIGHT BE BEST. That is, I will raise pre-flop with a high pair or two high cards.
If three low cards of different suits came on the flop, I would make a continuation bet if I had raised pre-flop. If it is with an overpair, my hand is probably still best (unless someone has two pair or a set, possible, but unlikely on such a flop).
If I had A-K, it's not clear if my hand were best or not, because someone might have paired. Still, I would make a continuation bet. Because if I made such bets with A-A or K-K post-flop, and not with A-K, people would some be able to tell the difference.
gonna go ahead and say it all depends....
If you're fairly sure you have the best hand you should c-bet boards that are likely to have hit your opponent so that he will call you. betting monotone/connected boards will likely make your opponent fold unless he has a draw to beat you.
If you're fairly sure that he has the best hand you should bet scary boards and hope he will fold.