All visitors to Martinique must have a valid passport (or a valid national identity card if you’re an EU citizen). A round-trip or onward ticket is officially required of non-EU visitors.
Customs regulations are the same for Martinique as they are for France. There are no abnormal restrictions, though it's important to note that anything made from coral or turtle shell is illegal and will confiscated if found in your bags.
Citizens of the USA, Canada, Australia and New Zealand can stay for up to 90 days without a visa, by showing a valid passport. Citizens of the EU can stay in Martinique indefinitely, and just need an official national identity card or a valid passport to enter the country.