The only shop worth mentioning in Nauru is Chappelle's in Ewa district. There is a threadbare supermarket in Aiwo which sells almost exclusively Black & Gold (budget Aussie brand) stock and there are countless small kiosks and shacks offering snacks and drinks, but Chappelle's is the only place to go for anything else, namely fresh produce, a decent range of food and drink, electronic goods, DIY stuff, clothing and material, homewares and a limited range of books. It's also the only shop that sells alcohol and the only place to buy a souvenir, since the duty free store in the airport is now closed.
What to buy: As above, buy fresh produce and booze here. If you're looking for gifts, there are some Nauru-branded clothes as well as other tropical shirts and sarongs etc.