In J. R. R. Tolkien's legendarium, the Elves are a sundered (divided) people. They awoke at Cuiviénen on the continent of Middle-earth (see: Awakening of the Elves), where they were divided into three tribes: Minyar (the Firsts), Tatyar (the Seconds) and Nelyar (the Thirds). After some time, they were summoned by Oromë to live with the Valar in Aman. That summoning and the Great Journey that followed split the Elves into two main groups (and many minor ones), which were never fully reunited.
The name Quendi refers to all elves.
The Avari are Those who Refused the Summoning of Oromë, or simply The Unwilling. Half of the Avari came from the largest tribe, the Nelyar, but most of the Nelyar went on the journey. Half of the Tatyar remained as well.
After the Separation the Avari became divided even more than the Eldar, though little of their history became known to the Elves and Men of the West of Middle-earth. At least six kindreds existed, and they continued to call themselves 'Quendi', considering those who went away as deserters. Some of these tribes later also journeyed westward, intermingling with the Nandor, and a few even reached Beleriand, though usually remaining in unfriendly relationship with the Sindar.
The Eldar are those who accepted the summons. Their name, literally Star People, was given to them by Oromë in their own language.