When you make a continuation bet after raising pre-flop you are looking to collect the dead money and build the pot for the times that you have a strong hand. There are a number of factors that come into play when making the decision whether to C-bet or not and many of these have to do with how your opponent plays different board textures.
Continuation betting on a dry flop will generally have the effect of eliminating players with speculative hands such as middle pairs and pocket pairs. It will also chase away timid players that are slow to call. A good rule of thumb is to bet two-thirds the size of the flop bet. This will be more than enough to collect the dead money and start building the pot for when you do have a strong hand.
However, there are a few situations where it is better to check instead of c-betting. For example, if you raise preflop and your opponent calls the flop and it is T 5 3 you are likely facing an aggressive player who will call with pocket aces, kings, queens, and jacks. They will even three bet with ace-king against you if they have a good hand. This type of player will be able to put you on a bluff because they will know that your C-bet is unlikely to be called.
In general, a C-bet should be automatic when you flop a set or two pair. This is especially true if you are in position, as you will be able to extract the maximum value from your strong hand. However, you should be less inclined to C-bet if the flop is a rainbow board with an ace, as it will give your opponents some cover and disguise and may encourage them to slow play a weaker hand.
You should also be less likely to c-bet on a draw-y board. These types of boards will almost always give your opponent something to call with and they will be able to connect more often. Passive players tend to overfold to a c-bet and so you should be more likely to check on these types of boards.
If you have a solid understanding of your opponents and their ranges it should be easy to determine what to do in each situation. The most important factor in deciding whether to c-bet or not is how well the board interacts with your opponents’ ranges. This can be easily analyzed using poker calculators like Poker Equilab.
Another important factor is the relative strength of your opponent’s range. Tight-passive players are much more likely to call your c-bet with a strong hand than a loose-passive player. Therefore, you should c-bet more frequently against tight-passive players. Conversely, you should be more inclined to check if you are up against a loose-passive player as they will rarely fold if you miss the flop. This makes them good candidates to fire into and to bet the turn and river cautiously if you are bluffing and aggressively when you have a strong hand.