Suppose your playing any type of poker, free online poker or cash and you have A-10 and the board comes up 8-6-Q-3-2. What hands do you expect your opponent to have? It may be Q-x, 7-6 or something.
However, notice the board: It contains no opportunities for Straights to form. So you can figure out that if your opponent called (just called) your last two bets which are bluffs, then you can put him on a Straight draw (possibly 9-7 or 10-9).
Now you want your opponent to believe that you have a Queen, but you feel that you have the best hand (and let us assume that you really do have the best hand). Let us add that you won't call a bet, because you have nothing, and you won't raise a bettor. Should you still bet, or just check?
Let us see the differences between the outcomes of these two actions. Suppose you are first to act, and you just check. If you just check, are you quite sure that your opponent will reveal his busted Straight draw? No.
If he has a hand as good as J-8 or 9-6 expect a showdown. But if your opponent missed his draws, then he will try to save himself the embarrassment of being a draw-chaser by bluffing, and you will fold.
He won't check Nine-high or Ten-high, of course. Expect a small bet which you won't call.
If your opponent is first to act and checks, and you check, it is with the conviction that your Ace-high is the best hand. However, revealing Ace-high will cement your reputation as a bluffer, and if you repeat bluffing later, others will be running you down with less than premium hands, such as second top Pair or even a small pocket Pair.
So you gain chips now, but at the cost of cramping your aggressive (maybe loose-aggressive) style.
Now suppose you are first to act, and you bet. Because your opponent has a busted draw, do not expect that he will call. You still win the pot. Furthermore, because the hand ended before you show down your hole cards, then your opponent will be left guessing as to what your hand really was.
This is the position you want to put your opponent in - he has a lot of guesswork to do, and his brain will be muddled as to what you are holding and as to what you may be holding if you repeat your bluffing later.
Even someone with 9-8 may fold. Why? Because you played strongly in the Flop and the Turn - and still in the river. Your opponent might put you on the Queen or on an overpair or eve J-J; in either case, he thinks his second top Pair is beat.
If your opponent checks, and you bet, it is almost the same as if you are first to act, and you bet.
Bluffing with the best hand is nearly a contradiction in terms: you bluff only if you make better hands fold. But, in this case, it may be better to make anybody else fold - best hand or worst hand - so as to preserve your unreadability which will pay off later.