In general, when we have a hand that beats a significant portion of our opponent's range, we will try to show it down. However, there are situations where we can make more profit both with our hand and other parts of our range by bluffing instead.
Consider the following hand:
In a HU $2/$4 match, the button raises and we call in the BB with 45. The flop comes down ajj, we check call, the turn is the 8, and it checks through. The river is the k.
Our range for check calling the flop might look something like any pair, any K high, and any gutshot (QT, KT and KQ since we aren't calling preflop with 53, 52 or 32). In this situation every hand that check calls the flop has at least a pair. Therefore, if he expects us to check our our weak pairs, and bet with our good pairs, two pairs and straights, he will fold some strongish bluff catchers, such as Tx. Therefore, we cannot really bet a hand even as strong as K9 for value. By turning some weak pairs into bluffs here, we can make a profit by bluffing him off some better pairs. Moreover, as he starts to adjust and call with more of these hands, we can extract more value from our Kx type hands.
What's happening in this hand is that our range here is unbalanced. Because our flop floats all hit hands on the river, we no longer have air in our range. Therefore we must create some hands to bluff with in order to balance our range. This play depends on your opponent's ability to hand read and correctly identify your range as unbalanced (as having very few bluffs). Against an opponent who is more likely just playing his hand, and who will bluff catch in these situations anyway, we can make more money by remaining unbalanced (value-betting thinly, but rarely bluffing).
A Different Example:
Now suppose we have 78, we raise to $12, and the BB calls. The flop comes down 973, we bet $18 into $24, the BB raises to $58, and we call. The turn is the 10, and the BB checks. The 10 improves so much of the BB's semi-bluffing range that we should expect him to continue a bluff here very often, and since he should not expect us to bet here very often, it is unlikely he will attempt a check raise with a big hand. Therefore his range looks a lot like a T (such as JT or T8), a strong 9, or a weak two pair that decides not to valuebet.. Our range looks fairly strong, since we can have flushes, a straight, some top pairs, some overpairs and some two pairs. Most of our range here is at least a pair, since the 10s on the turn completes or pairs the majority of our draws. Unless our opponent is a weaker player (who can't read hands or fold medium strength hands) or thinks we are turning hands into bluffs, we can bet twice to bluff him off of better pairs. Again it is important to know your opponent, since some (generally weaker) villains will check call here with surprisingly strong hands (maybe even sets), get very tricky and check a monster, or talk themselves into calling here with bluff-catchers.
We call with a5 in the BB, and check call a k75 flop. The 7 falls on the turn and it checks through. The river is the 10.
When the button checks the turn, he is frequently giving up, since most players perceive the 7 to be a bad card to bluff. Our hand is too weak to value-bet, so we check hoping to show it down and very often win. In this situation we would probably bluff with our air, since we expect the button to be giving up a lot, so our range looks very showdownable. Therefore, we generally expect button to bluff here infrequently, and to go for thin value a lot (probably Tx at least). This makes this look like a good spot to check raise bluff, and in some sense it is. It is true that we don't have a lot of air in our range, and it is also true that his range is not very strong, but ours isn't either. While it might be a good play to check raise here with our 7x and maybe some of our Kx, we also will want to bet those hands out to balance the times we are bluffing. We need to distribute these value hands between our two ranges, which can be difficult, and if we make this play too often or don't bet out for value enough, a good opponent will sniff it out and we'll end up getting called a lot when our range is weak.
When should we make this play?
In general, we should consider these plays when our range looks like it doesn't have a lot of air, but does have some strong hands, and our opponent's range is capped or weak. It works best against opponents who can recognize this, but who won't expect us to turn hands into bluffs. However, these plays can be overused, since the part of our range we can turn into a bluff is often pretty big, and a good opponent will exploit this.