Iceland is a member of the Schengen Agreement, according to which travellers from 26 EU countries are extempt from border controls. Residents of a Schengen country can stay in Iceland for 90 days without applying for a visa.
Residents outside the Schengen area need to present a valid passport (valid for at least three months beyond the date of entry). Here you can find a list of countries whose residents need a visa to enter Iceland.