When two American adults meet for the first time, or when they know each other but not well enough to hug, they usually put their right hands together. Especially if they're men. As a handbook on American customs posted on a University of Texas-Arlington web site says, "Some men might not shake hands with women unless the woman extends her hand first. Hand-shaking among women occurs even less frequently."
Now, here's the rest of what that handbook for foreign visitors says about how "we" shake hands--is this really how all of us shake hands?
Obviously, many African American men in particular have other ways of putting their hands together, and other racial groups do as well (though I'll admit, I don't know what forms the latter take). So this visitor's handbook may be explaining the "normal" American method, but it's really the "white" method. Adopting it, and dropping any other "ethnic" greeting gestures, has been another way in which immigrants have adapted in order to assimilate. Shaking hands this way is a custom that originated in Europe, perhaps as a way of showing that the hand doesn't contain a weapon (this would also explain why the right hand is always used). In the U.S., it's specific European origins became whitewashed into an "American" custom, and since, until the 20th Century, the real Americans were openly labeled "white Anglo-Saxon" Americans, the standard handshake became a white custom.
When Americans shake hands, they normally exert a small amount of pressure on each others' hands, move their clasped hands a bit upwards, then a bit downwards, then release their grip. People from other places where handshaking is customary may hold the other person's hand more or less firmly than Americans do, sustaining contact for a shorter or a longer time than Americans. One's character in the U.S. is often assumed by the appropriateness of their handshake.
What's more interesting, though, about differences in handshaking techniques is that if a white and a non-white person encounter each other in a casual setting and decide to clasp hands, there may be uncertainty about which handshaking method to use--the one that's become the standard, "white" one, or a common non-white one. When there is uncertainty about which to use, the fall-back is usually the standard handshake, that is, the method more likely to be used by the white person than by the one used by some non-white people. The non-white person often represses a preferred method of contact, and the white person feels little if any discomfort about being the enforcer of a standard.