There is a lot more to the story than is told in the The Lord of the Rings (which is, essentially, the tale of the end of the Third Age and the beginning of the Fourth Age - the Age of Man). You must also read the Silmarillion and several of the other books to acquire the full story.
So when the Elves sail away to the Gray Havens, actually, they vanished? Or became Eskimos, American Indians, or perhaps characters from the Book of Mormon?
They passed through the Grey Havens to the "Blessed Realms" of Aman, the Undying Lands. These were the lands to the west of the Great Sea, including Valinor, the land of the Valar. After the Downfall of N?menor, they were taken away from the mortal realms, and could only be reached by those permitted to travel the Straight Road (elves, mostly). This was a realm that was between the mortal realms of Arda (Earth) and the Timeless Halls of Eru Il?vatar and the Ainur (essentially Heaven).
I did wonder why Tom Bombadil somehow vanished from the story in the movie version, which was otherwise pretty faithful to the trilogy.
There were lots of other changes. From the first movie alone: Saruman tells Gandalf that Sauron cannot take physical form, yet in the book he already had. Aragorn gave them the weapons at Bree instead of the Hobbits finding them in the Barrow Downs. Glorfindel is gone, replaced by an expanded role for Arwen. Also, in the books Arwen never carries a weapon, yet in the movies she seems to be a master of swords. The movie tells us that Aragorn renounced his kingship, yet in the books he never did. In the book there are five wizards and one of the others (Radagast) sends the eagle to save Gandalf - in the movie Gandalf used a moth to summon the eagle. The book has Sauron causing their difficulties in attempting the Redhorn Pass, while in the movie it was Saruman. Lurtz, the Uruk chieftain, does not appear in the book.
The maps in the LOTR do not appear to correspond to Europe or the British Isles.
In the language of the Elder Days, 'Arda' signified the World and all that is in it. Arda was created through the Music of the Ainur to be a dwelling place for the Children of Il?vatar (that is, Elves and Men).
Originally a flat world, the continents were surrounded by a mighty ocean, Ekkaia or the Encircling Sea, and separated by Belegaer, called the Great Sea. In the First Age, the regions to the north and west of Middle-earth was occupied by the country of Beleriand, but this was destroyed during the tumults of the War of Wrath.
In the Second Age, the isle of N?menor was raised in the Great Sea for the Edain to dwell on. This island existed through most of the Second Age, but was destroyed in the Downfall of N?menor as a result of the pride of the N?menorean people in defying the Ban of the Valar and sailing to Aman in the west.
After the Downfall, the shape of Arda was changed, and it was made 'bent' (or 'round' as we should say in modern terms). Aman was taken out of the world, and could only be reached by the Elves, following the straight road that was granted to them. As Aman was taken away from Arda, new lands and continents were created. From the late Second Age onwards, the World began to take the shape that we know today.