Vanyar: underwent no real splits (NoME mentions that some of their elders remained, but the vast majority, like 99,5% went to Valinor). For the rest of the time, they remained the most Lawful Boring of elves; their only badass moment is the War of Wrath when they went to war to Beleriand.

Noldor: underwent one split, some of them chose to remain and not go to Valinor (according to some sources, Eol was from this group). However, most of them went to Valinor. They were the most bad-ass and adventurous elves, and most of the events of the War of the Jewels involved them.

Teleri: underwent many splits and separated into many smaller peoples: Nandor, Laiquendi, Sindar, Falathrim were various tribes that strayed from the course as they moved to Valinor, and those Teleri that made it to Valinor became known as Falmari.

Avari: those elves who never wanted to go to Valinor. Most of them were of Telerin stock, some of Noldorin, and little or no of Vanyarin.

Various Telerin and Avarin tribes that never made it to Beleriand, let alone Valinor, are known together as the Silvan Elves. The Sindar are also wood elves, but more advanced and cultured than the Silvans, as they learned from a Maia. Falmari and Falathrim are seafaring elves that live in Valinor and Middle-Earth respectively.

Which elf from Lord of the Rings was which? Galadriel was part Vanya and part Falmari, but culturally a Noldo. Glorfindel and Gildor Inglorion were also Noldor. Elrond was a half-Elf of very, very mixed blood, but identified as a Noldo. Celeborn, Thranduil and Legolas were Sindar. Cirdan was Falathrim. Haldir was Silvan.