Theres quite a few elven kingdoms actually that aren't either Lothlorien, Mirkwood, or Rivendell. Most I think don't exist at the time of the LoTR story, but this is what i found on it:
Elves of the Havens: A Sindarin people occupying the land of the Falas in western Beleriand, on the shores of Belegaer. Círdan the Shipwright was their lord, and their great cities were the Havens of Brithombar and Eglarest.
The Elves of the Havens enjoyed peaceful relations with all the Elves of Beleriand, and were particularly closely allied with the people of Nargothrond. As one of the most remote peoples from Morgoth's forces in Angband, the other Elvish lords would often send their children to the Elves of the Havens for safekeeping.
After the Nirnaeth Arnoediad, Brithombar and Eglarest were besieged and taken by Morgoth's armies, and most of the Elves were slain or imprisoned. Some few, including Círdan and the young Ereinion Gil-galad, who was at that time lodging with the people of the Havens, sailed south to safety on the Isle of Balar.
At the beginning of the Second Age, Círdan sailed east to found the Grey Havens on the Gulf of Lhûn, and the remnant of the Elves of the Havens must surely have accompanied him.
Elves of the Falas: In ancient days, the Telerin Elves dwelt for long years in Beleriand, and they learned much from Ossë, a Maia of the Sea. When Ossë's master Ulmo conducted the most part of the Teleri into the West, Ossë persuaded some few to remain behind. They settled on Beleriand's western shores, building themselves Havens at Brithombar and Eglarest. Círdan was their leader, and they were known as the Falathrim, or Elves of the Falas, the first builders and sailors of ships in Middle-earth's history.
After the Return of the Noldor, the Elves of the Falas became fast in friendship with the Deep Elves. With Finrod's aid, the Havens of the Falas were strengthened and rebuilt. They fought beside the Noldor, too: in the Dagor Bragollach, their mariners rescued High King Fingon from overwhelming hordes in Hithlum, and they went to the aid of Fingon again at the Nirnaeth Arnoediad.
After the disaster of the Nirnaeth, many of the survivors fled back to the Havens of the Falas, but they were not safe there for long. A year after the great defeat, Morgoth sent his creatures to besiege them, and both the Havens were overthrown. Most of the Elves of the Falas perished in the fall of the Havens, or were captured, but Círdan escaped. With his surviving followers, he sailed southwards, and settled for a time on the Isle of Balar. He maintained a haven there for many years, before removing once again to Lindon, where he dwelt at the Grey Havens with, doubtless, a remnant of the Falathrim.
Elves of lindon: Those Elves who dwelt in the west of Middle-earth, between the Blue Mountains and the Great Sea, in the Second and Third Ages. They reached the height of their power in the Second Age under King Gil-galad, but waned after his loss in the Siege of Barad-dûr.
There are also different branches of the race of elves, like the Avari and the Teleri.
Avari: 'The Unwilling'; those of the Elves who refused the summons of the Valar to dwell in Valinor, and remained in the dark wilds of Middle-earth. Some of the Avari, it is said, were corrupted by Melkor in ancient days to become the progenitors of the race of Orcs.
Teleri: The third and greatest of the three Elf-kindreds of the Great Journey, from whom the Nandor and the Sindar were descended. The Teleri came last to Aman, and dwelt in Alqualondë on the western shores of the Great Sea.
All info found at
www.glyphweb.com/arda/default.htm?http://www.glyphweb.com/arda/n/noldor.html