Over the past two centuries, this tiny plot of land has been known as the Kingdom of Larry, the Republic of Stevie-Ray, Billsylvania, the New Republic of Stevie-Ray, the Kingdom of Tom, and finally the Republic of Dave after Dave overthrew his father.
The Republic does not accept new citizens; however, one may apply for asylum (the right to stay temporarily). For this you must consult the leader, Dave, and tell him that you are looking for asylum, or for a "special asylum" you can become a "refugee" within the country. The standard fee, according to Dave, is 250 caps which presumably goes towards funding the administration.
The Republic of Dave is one of, if not the only, areas of the Capital Wasteland with a democratic government, probably modeled after pre-war America. The Republic is largely self-sufficient in the fact that it raises it's own brahmin; however, if you are willing to abide by their rules, and are not trying to "spy" on them, the residents of the Republic are usually more than happy to trade with friendly wastelanders.
For a list of destinations the Republic of Dave has to offer, see Republic of Dave: Tourist Information