While Greenland, Madagascar and Tasmania rifted away from other Continental land masses, the great majority of oceanic islands are volcanic in origin. Of these, most lie along subduction zones, where one of Earth's tectonic plates is subducting beneath another; in these areas, the crust of the subducted plate begins to melt as it approaches the mantle, producing a line of volcanoes on the overlying plate.
The Aleutians and northern islands of Japan have formed where the Pacific Plate is subducting beneath the North American Plate; in like manner, the archipelagos of Indonesia, southeast Asia (Taiwan, the Philippines, southern Japan) and the South Pacific (Solomon Islands, New Zealand) rise above subduction zones at the margins of the Pacific, Australian and Philippine Plates. The Lesser Antilles, stretching along the eastern edge of the Caribbean Sea, were produced by subduction of both the North and South American Plates beneath the Caribbean Plate.
Iceland, which rose from the sea 16 million years ago, has resulted from volcanism along the mid Atlantic Ridge (where the spreading Atlantic continues to rift North America from Eurasia); indeed, this island nation straddles the two Continental Plates. A third type of volcanic island, produced by a mantle plume, can develop far from any plate margin; the Hawaiian Island chain, the Galapagos Islands, the Azores and Easter Island are examples of such isolated "hotspot" volcanism.
Of course, all of these islands are prone to continued volcanic activity, even after hundreds or thousands of years of quiescence; only hotspot islands that have been transported away from the mantle plume (by movement of their oceanic plate) are immune to reawakening. Earth's landscape continues to evolve and humans are finally around to observe it!