The play takes place entirely on an island (and the water surrounding it). We have a confession, Shmoopsters. We're not exactly sure where the island is located. Since the King of Naples and his party were travelling between Naples and Tunis when Prospero whipped up the little storm that shipwrecked them, it seems likely the island is located somewhere in the Mediterranean.
Yet, there are some obvious references to the Caribbean, which has led some scholars to suggest the island is either located in the New World or that the New World was on Shakespeare's mind when he wrote the play. Ariel's reference to "the still-vex'd Bermoothes" is a pretty clear shout-out to Bermuda, previously avoided because it was thought to be the devil's island.
The Caribbean isles might also have been fresh in Shakespeare's imagination because of a popular travel account written around the same times as the play: In 1609, a Virginia Company expedition to Jamestown lost its flagship, the Sea Venture. It was assumed that the flagship was as good as gone, but then a year later, the crew turned up happy and tan, after having passed the time in Bermuda. The account of that adventure was published as A Discovery of the Bermudas, Otherwise Called the Isle of the Devils and was a very popularly read. (Learn more about the Sea Venture shipwreck here.) Did we mention that Caliban's name is probably an anagram of the word "can[n]ibal," a term derived from "carib" (as in the Caribbean)?
What's important is that the place is an island, and thus removed from territorial or national claims, and, most importantly, any kind of civilization – including the King's court and Prospero's ex-dukedom in Milan. If you want to know more about the setting, check the "Themes: Contrasting Regions."
Brain Snack: The fancy setting also just so happens to work out for Princess Elizabeth, daughter of the English King James I, who enjoyed a special production of the play as part of her wedding festivities. The whimsical setting probably made for a pretty magical evening.