Well, briefly:
The Valar, along with the Maiar, enter Arda (the world) and begin creating the physical expression of the song of the Ainur, before the world was made. In the center of the realm they build two lamps, on high towers to light the world. Melkor begins perverting everything they do. The struggle is so titanic that much of the early beauty of Arda is marred forever, especially those things deep in the earth, and in the far north, where fire, and ice are exploited by Melkor. In the end, the Valar have their first home destroyed by Melkor, when he throws down the towers of the lanterns, and causes the earth to be broken. The Valar end up chaining Melkor up in Mandos for a very long time. Eventually they let him out, and he plays nice for a while, and then slips off alone while no one is paying attention. (Evidently, the Valar are not big on paying attention to things for long periods.)
The world has been broken, and the Valar decide to move into the western lands, over the sea. There they establish the realm of Aman, and bless it. They grow two trees there, which provide, in their flowers and fruits a new source of light for the world. The light drips from the trees, and is gathered up in pools. The lands of the west remain in darkness. Elbereth has pity upon the Children of Iluvatar who are prophesied to come into Arda in the eastern lands. (Called by the Valar, the Quendi, meaning those who speak) She takes up some of the light and splashes it out into the sky, making stars to light the darkness for the Elves, when they come. For this she is revered greatly by the Elves.
Orome, one of the Valar meets three kindreds of elves in Middle Earth, while on a hunting trip. He brings their leaders back to Aman with him, to see the blessed Realm, and meet with the rest of the Valar, and Maiar. The are Ingwe, Finwe, and Elwe, The kindreds are the Vanyar, Noldor, and Teleri. The leaders return to Middle earth and lead their people back to Aman. All the ones who made it back and saw the light of the two trees were the Caliquendi, the elves of the light. The ones who did not go were the Morequendi, the elves of darkness.
The three kindreds did not travel together, and only the Vanyar went directly to Aman, and remained. The Noldor went a bit more slowly, and eventually a great number of them returned, for reasons important to the history of elves, and men. The Teleri took their sweet time going, got split up, when their king wandered off into the woods and ended up holding hands with a Maia named Melian for long enough that the trees grew tall, while they just stood there. (OK, I take back the part about not paying attention to things for a long time.) So, half the Teleri remained behind, and the rest set off for Aman on their own with the first kings brother. They never really quite made it, though, and were given an island just off the coast of Aman, and some of the land on its outside coast to live on. They did see the light of the trees, though, and are considered Caliquendi. They also made some nifty boats.
The elven king who stayed behind, Elwe Singollo, was later known as Elu Thingol, and he established a hidden kingdom in Middle Earth. Melian the Maia established an enchantment that hid it, and prevented anything from entering it, unless Thingol wanted it to.
Not long after this (well, not long in elvish terms) Melkor comes back, with a nasty friend, and kills the two trees, kills the king of the Noldor, and steals the prized possession of Feanor, the badest assed elf of all time. (Except maybe Fingolfin, depending on how you view such things.) Feanor renames Melkor Morgoth, and speaks the dreaded Oath of Feanor. Much sorrow ensues, including yet another war by the Valar, and yet another land or two dropped into the depths of the sea. (note: Do not fuck with the Valar.)
The elves came back to Middle Earth in order not to miss out on all the fun. The Teleri got used, and some of them killed during this, as did a goodly number of the Noldor. Men end up being involved, and also got to die in large numbers as well. It all ends with Morgoth being kicked out of Arda entirely, and the Valar letting the Noldor come back to Aman if they want. Several thousand years later, the last of them do so, bringing with them a Hobbit, of all things.
I had to leave out a lot of the good stuff, in order to explain the going west thing.
Tris