Havelock Island is the most visited of the Andaman Islands. It's becoming popular for its beaches, laid back atmoshere and great scuba diving.
The Indian government has been advocating eco-friendly tourism development on Havelock for a while now, and while it's the most visited of the islands it sees only a fraction of the visitors that crowd other island destinations such as Thailand. The pace of life is slow, the locals are friendly and still not too affected by tourism, and the facilities are moderately basic for the most part.
Havelock is the main center for diving in the island chain. Usual places for diving are Elephant beach and Lighthouse in Havelock. If you're an experienced diver, also enquire about dive expeditions, where they will take you out hunting for new and undiscovered sites.