For being such a small island, Rapa Nui offers a surprisingly good nightlife.
Topatangi on the main street of Atamu Tekena is a very popular pub. It's open at least two days a week - thursday and saturday - and opens at 23:00. Here you'll find people of all ages. On thursdays, the band of the same name as the pub offers free shows with male dancers for a few songs, which attracts a big audience.
Easter Island has two discos. One is Piriti, which is a bit outside of the town center along the street Hotu Matu'a - the same street that also passes the airport. When this club closes, those who are still going strong may continue to the disco Toroko. This one is on the opposite side of the town center. Toroko offers a "rougher" ambient, where you'll be able to party until sunrise.
Other than these places, there are several restaurants and pubs along the main street of Atamu Tekena that offer alcoholic beverages.