Reverse psychology is a not-very-well-agreed-upon phrase which generally refers to compensating for what one believes one's adversary wants, by saying the opposite of what that person expects, or doing the opposite of what that person asks.
The term is used very loosely in a number of ways, but a common form is saying the opposite of what one actually wants, knowing that the other party, being hostile, will do the opposite, and hence will actually do what one wants.
A common use of this shows up in the "Tar baby" story of the classic Uncle Remus stories by Joel Chandler Harris. The three characters are Br'er Bear, Br'er Fox, and Br'er Rabbit. The first two of these are allies of each other and enemies of Br'er Rabbit. They have caught Br'er Rabbit because of an unfortunate incident involving a Tar baby and are discussing ways of killing him. Br'er Rabbit has no objection to any of the methods (skinning, hanging, etc.), but requests that, whatever they do, they don't throw him in the briar patch. Br'er Bear and Br'er Fox of course do exactly that. It then turns out that the briar patch is Br'er Rabbit's natural home.
Of course, this use of "reverse psychology" can be overdone. One might compensate for what one's adversary is doing, and reverse things again. In effect, one might think "Br'er Rabbit says he doesn't want to be thrown in the briar patch, but I've read the Uncle Remus stories, and know that he is bluffing. So I will do the opposite of what he asks."
Of course, this can go on and on, until the term is really rather meaningless.
- The stories are told in a not-very-literate slave dialect, and "Br'er" means "brother"