The Avari are a branch of the Elves.
When Oromë found the Elves that awakened in Cuivienen (see: Awakening of the Elves), he summoned them to come with him to Valinor. All the Vanyar and most of the Ñoldor were persuaded, along with some of the Teleri, and followed Oromë into the west on the Great Journey. The remainder of the Ñoldor and Teleri remained suspicious, or simply refused to depart from their own lands, and spread gradually throughout the wide lands of Middle-earth. They were after known in Quenya—the language of the Eldar that eventually reached Valinor—by the name Avari, meaning "the Unwilling", because they refused the summons.
Having never come to Valinor, the Avari remained a wild folk, dwellers of forests. Little is known of them, as they do not appear in any of the tales, save some references to Avari creeping in the south of Beleriand in the First Age. Many of them probably merged with the Nandor and became known as Silvan Elves. Also, it is speculated that the Dark Elves were the first other sentient race encountered by the race of Men during their infancy. The Dark Elves probably taught them many of the basic crafts of civilization, though the craft of the Eldar surpassed that of the Avari even more than that of the Avari surpassed primitive Men.
In The War of the Jewels, names of six tribes of Avari in their own languages are given, all being cognates of the Quenya word Quendi (the Speakers): Kindi, Cuind, Hwenti, Windan, Kinn-lai, Penni. They are the only certain Avarin words ever mentioned in the published Middle-earth material. It is speculated however that Dorwinion was an Avarin land, with Winion carrying the meaning of "Wine".
Other versions of the legendarium
In older versions of the legendarium, the name Avari was originally that of the later Eldar, then meaning "those that departed".
In other, relatively late writings, a brief idea was that the Avari did not come from the three clans, but from two other clans, led by Nurwë and Morwë. This idea was later dropped. In the final conception, the Elves were divided into three tribes.