The dap is a ritualistic greeting between two persons that involves a sequence of touching, gripping and bumping of hands with the other person.
This custom developed among black Americans, most likely as an elaboration of the sliding handshake ('slip me some skin, brother', 'slip me five', 'give me five,' and later the 'high five', etc.), which originated as a show of appreciation or mutual recognition and understanding among black jazz and swing musicians of earlier decades.
In the 60's, the dap became an exclusive display of black identity, pride, solidarity and brotherhood, particularly among young black soldiers serving in Vietnam. In that era, the sequence became extended and elaborate, sometimes taking dozens of seconds to complete. A dap would be initiated by one man raising a fist. The other man would hold out a fist for the first man to hit in a light downward pounding motion. Then the roles reversed and the action repeated. After the initial fist pounding, the dap could branch off into a variety of hand clasps, slides, locking of thumbs, sideways tapping and other motions that were always executed symmetrically and with a smooth grace. Groups developed signature daps. One Viet Nam vet mentions an abbreviated 'Ti-ti dap' (ti-ti means small in Vietnamese) for close friends and a 'rabbit dap' for white guys that had their shit together.
The origin of the term is not clear, but the Internet offers some speculation. One is a possible back-formation of the acronym for Dignity and Pride. Another is similarity to the Vietnamese word for 'beautiful', supposedly because the Vietnamese who witnessed the dap were impressed by the display.
The dap did not disappear in the decades after the Viet Nam war, but it had faded from popular black culture enough to reflect a generation gap a decade or two later. One Viet Nam war veteran related in the 80's, "My teen kids have adopted rather elaborate handshakes...it's happenin Dad! They however, refuse to believe that it's been done before, and much more elaborately."
Now, nearly half a century later, the dap is alive and well in our youth culture and street culture. The best way to get it is to watch people do it:
Jane and the pygmy (sadly, this image seems to have disappeared from the Web. It is a scene from the 1936 film of Tarzan Escapes in which Jane greets a pygmy man with a fist bump.)
'On the street'
See also: fist bump and pound hug.
A note on the ephemeral Web and fading history:
When I first wrote this, some years ago, there was much more info available on the Web, especially in informal 'blog' type relations of the personal experiences of Vietnam war vets, all of which Google seems unable to find now, so I was unable to expand on or update the topic, and some of the original links are now dead-ends.