What are all the different types of Elves?

At the beginning all Elves were counted as Eldar. The original division of the Elves broke down into those Eldar who took part in the call of the Valar to make the Great Journey to Valinor and the Avari who did not fully trust the Valar and refused. The Avari are commonly known as East-elves.

The Eldar of the Great Journey was comprised of three groups:

1. The Vanyar were the first to set off for the West and the first to reach the lands of Beleriand. The Vanyar were recogized by their golden hair and the white banners they bore.

2. The second of the Three Kindreds of the Eldar was the Noldor or the High Elves. They sailed to Eldamar with the Vanyar but grew restless and rebellious listening to the lies of Melkor. The Noldor had dark hair and grey eyes except for the golden-haired House of Finarfin.

3. Finally there was the Teleri, sometimes called the Sea-elves, many of whom completed the Great Journey, but stopped either east of the Misty Mountains or at the shores of Beleriand. There were many lesser divisions among the Teleri such as the Falmari, the Sindar and the Nandor. Some of the Nandor mixed with Avari who came to Rhovanion later, settling in the Greenwood and Laurelindórenan (later Lothlórien) as the Silvan Elves.

https://www.youtube.com/watch?v=43Wy3STjXnc

Warning: Spoilers included if you haven't finished The Silmarillion.

The Moriquendi weren't evil like the Drow are. They are called Dark Elves because they chose to remain in Middle-earth, which was covered in darkness until the Sun's first rising, rather than travelling to Aman, which was lit by the Two Trees.

Though it says in the chapter of the Sindar, that they are called the starlit Elves of Beleriand and though they were of the Moriquendi they became under the teachings of Melian and with the guidance of Thingol the wisest and the fairest of all Elves of Middle Earth!