A travel guide I've read says that Barbuda is "one of the very few Leeward Islands that Columbus did not discover". It is unclear to me how one could discover Antigua and not Barbuda, but maybe the visibility was bad and he was in a hurry.
Barbuda is practically a wildlife resort because the population density is so low.
There are only a few hotels/resorts (the late Princess of Wales is said to have
spent some time there). The long, sandy beaches are protected by reefs which
are littered by "a few centuries' worth of shipwrecks" (quoting from the marketing
brochure from which I also took the left image below).
Barbuda's beach and airport terminal building. Download full size (1280x960) left, right.
There really are no attractions of note on Barbuda - you go there for a quiet and undisturbed swim (or snorkelling, or sunbathing) - although the occasional ruin bears witness to past fighting. The highest point in the "North-Eastern Highlands" is a majestic 125 feet, but there's also a natural cave 400 feet under ground, and several sink holes of interest to tourists and botanists alike - a few plant species are reportedly world-exclusive to this little island!
Fire bridgade at the airport, and goats feasting on a hedge. Full size (1280x960) left, right.
The most prominent inhabitant of Barbuda is the Frigate Bird; the island is home to the largest colony in the world. There's a Nature Reserve in a lagoon not far from the airport, and while there's not really a "scheduled" boat service, local fishermen are used to ferrying tourists around that for a fixed price (which, as far as I remember, is rather high - in the region of US$ 20 per person).
There is a daily flight to Barbuda from Antigua, so you can go there on your own, but be sure to book ahead because there are less than 20 seats on the plane. The travel time is about 15 minutes. There's no scheduled boat service but I guess a few US dollars will easily persuade a fisherman to take you there if you want; that would then probably be a journey of about 2 hours. - If you wander into the village from the airport, you'll find a little café, and you can ask around for transport and/or accommodation.
Package deals are also available from local travel agents.