The islands that we observe in Earth's oceans today were formed by one of three processes: flooding, volcanism or rifting; in some cases, more than one process was involved. Most of the islands near Continental coasts are merely high ground on the continental shelf, cut off from the mainland when sea level rose after the Pleistocene glaciers retreated; Great Britain, Ireland, Newfoundland, Long Island, Sri Lanka and Tasmania are all examples of such islands.
Most oceanic islands are volcanic in origin, having developed in one of three ways. Some develop by volcanism along a spreading ridge, where plates are moving apart and new ocean crust is forming; Iceland and the Azores are prime examples (the Azores occur at a site where three tectonic plates, North American, Eurasian and African, are pulling apart). Hotspot islands, such as the Hawaiian, Galapagos and Cape Verde Islands have developed above a mantle plume, often far from a plate margin. Finally, volcanic island chains form along subduction zones where one plate is being forced beneath another; the Aleutians, Japan, outer Indonesian islands, the Philippines, the Solomon Islands, New Zealand and the eastern Caribbean chain are all examples of subduction island arcs. For every volcanic island that breaks the surface, there are hundreds of "seamounts" that stopped forming at an earlier stage.
The rifting of small land masses from larger, continental cratons has been occurring throughout our planet's history. Madagascar is the classic example; having separated from Africa during the breakup of Pangea, it attached to both Antarctica and India before anchoring off Africa once again. The Falkland Islands, once a piece of South Africa, moved off with South America as the southern Atlantic opened and Vancouver Island, once a slice of Australia, is now prime Canadian real estate. And, in the "near future" (geologically speaking), Southern California and the Baja will become an island in the Pacific.