Region: Middle Earth
The "island" is not an island and it is called Valinor
Long story short is that Valinor was another continent west of Middle Earth where the Valar (the "gods") lived some 10,000 years before the events of LOTR. Their big evil brother Morgoth was bullying everyone in Middle Earth, so the Valar brought the Elves (only those who wanted to go) to Valinor as a way of protecting them, hence the moniker "Undying Lands". Skipping over many, many events, and a long time later the men of Numenor are tricked into attacking Valinor by Sauron. Eru ("God") and the Valar decide enough is enough and move the continent into what is effectively outer space/off the planet and teach some Elves how to sail off the planet if they so wish to go there (which they all eventually do).
The Commonwealth of Lindon-Rivendell is right in saying that the ringbearers (Frodo, Bilbo, and Sam) had to go. Also, I think, it is also due in part to the fact that there is no "evil" (Sauron) left in Middle Earth except in those who carried the ring, so they must leave. Gandalf is from Valinor, and Elrond, Galadriel, and Cirdan (the Elf guy standing by the ship in movie) go because they are Elves and that is where they "belong".
I hope that clarifies things, though it is very condensed.