[lathspell]

Yes, The Elves were exiles.
It began when Feänor wrought the Silmarils and Melkor began to lust for them. He wanted but knew he could not get them by himself so he made a pact with Ungoliant to attack Valinor and steal the Silmarils and they did so. And with it they destroyed the Trees of Light, which Yavanna made.
After this the Elves were outraged, especially Feänor. He announced he was going to get the Silmarils back and made an Oath, the Oath of Feänor, saying that he would not rest before he had the Silmarils back and killing all those who would oppose him to do so. Many others, including his seven sons, made this vow as well, and where ensnared in Feänor's Doom. They wanted to take the Teleri ships to M-e, but they didn't let them take it. So Feänor and his people slew the Teleri and stole their ships. This is the Kinslaying of Alqualondë, and was most grievous of all Elf-deeds. Mandos came to them and said they still had one chance to return to Valinor and live at peace there and told them that if they would proceed with this journey there would be great grief and losses uncounted. They didn't listen to him and thus became ensnared in the Doom of Mandos and were exiles.

If you want to know the whole story, please read the Silmarillion in which it is all told neatly.

greetings,
lathspell