As with many things, there's a perfectly good Wikipedia article.
It's a club for a limited number of Yale seniors, who select their successors every spring. Like many of the other Yale senior societies, it was founded in the late 19th Century in imitation of similar clubs at German universities of the time. It is famous because it is rich, because it traditionally got a large share of BMOCs (and now BWOCs) many of whom went on to public prominence as adults (e.g., both George Bushes, John Kerrey, William F. Buckley, Jr., Strobe Talbott, Potter Stewart, James Jesus Angleton, Austan Goolsbee), and because it is particularly asinine and self-involved. Also, its windowless "tomb" sits smack in the middle of the Old Campus.
The membership has never been particularly secret -- it used to be listed in the yearbook -- but its internal workings and rituals are. Mostly, at least in the past, they seemed to consist of drinking and writing (and reading out loud) their own sexual autobiographies. Members are supposed to leave the room if anyone starts talking about it, which has been easy fodder for generations of jokes at Hasty Pudding shows and on 30 Rock.
I don't know what it is like now, but in the past at least it was fairly egalitarian (although women have been admitted only recently), with most of the members chosen because of actual accomplishments at the College rather than on the basis of wealth or social position in the outside world, and decent numbers of non-WASPs and people of color. Every year there are a few legacies who seem to have nothing but that to offer, though. It has a great alumni network, thanks in part to annual retreats on an island the society owns in the St. Lawrence River; the member I know best got a couple of great opportunities in early and mid-career that sure looked Bones-related.