The elves were created by Eru Ilúvatar (God), and they were the fairest and wisest of his children. When they awakened on the young Arda (earth) and learned to speak, they called themselves the Quendi (speakers).
Later there was a war between the Valar (demigods) and the evil Vala Melkor (Morgoth), and afterwards the Vala Oromë decided to take the elves with him to Aman, a continent in the west where the Valar's realm Valinor was located. Oromë loved the elves, and it was he who named them the Eldar (people of the stars).
Some elves wished to stay in Middle-earth; those where called the Avari, and later Moriquendi (dark elves) because they never got to see the light of the Trees in Valinor.
Those who followed Oromë were three clans: the Vanyar, the Noldor and the Teleri.
During the journey, Oromë's path was blocked by the Misty Mountains, and he went ahead to create a path through them. But when he finally returned, some of the Teleri had come to like the place so much they wanted to stay. Those were called the Nandor, and settled in Greenwood (which was renamed Mirkwood in the Third Age).
A few of he Nandor later went through the Misty Mountains after all, but settled right on the other side of it, those were called the Laiquendi (Green elves). Those who remained in Greenwood became known as the Silvan.
Now Oromë and the elves still following him had come into Beleriand, a country that lay west of the Misty Mountains and later got destroyed. Here they again lost some elves, for a Teleri leader disappeared into the forest of Doriath (his name was Elu Thingol and the reason was - of course - a pretty girl: the Maia Melian). Those who remained to wait for him settled in Doriath and became the Sindar (Grey elves).
Meanwhile, Oromë and the Vanyar and the Noldor went to the coast, where Ulmo, the Vala of the Seas awaited them and ferried them over the ocean to Aman. But the remaining Teleri (those who were not waiting for Thingol in Doriath) had been dragging their feet, probably talking to trees, singing etc., and so when they finally arrived, they found that Ulmo's ferry had already departed.
The Teleri waited for him, but when he returned to pick them up, part of them had fallen in love with the coast and decided to stay in Middle-earth after all. Those were called the Falathrim, and their leader was Círdan Shipwright. Those who went to Aman were called the Falmari. Both these Teleri branches became great ship builders, and it was because of the Falmari's ships, the Noldor later caused the first Kinslaying (with elves killing other elves) when they wanted to return to Middle-earth.
Just some research I did for my fanfiction and thought I might share. :)
