Eldar Corsairs (also known as Anhrathe) are the fifth major Eldar faction, after the Craftworlders, Harlequins, Exodites, and Dark Eldar. As the name implies, they are nomadic fleets of space pirates. They mostly consist of former Craftworld Eldar walking the path of the Outcast, although others are Dark Eldar who left Commorragh. They can be seen as a sort of half-way point between the two: hedonistic pirates, sure, but not absolute psychopathic sadists, and they use soulstones and psykers unlike the Commorrites.
The Corsairs are said to be the closest faction to the original Eldar before the Fall. Free from the bondage of the Path, they can explore the full extent of sensation in the universe without restraint. Their strong passions means that they are capable of both acts of great cruelty and kindness depending on their whims. Although they resort to piracy to survive due to lacking a economic support structure like a Craftworld or Kabal, it is mainly motivated by the fact that without the War Mask taught through the Path of the Warrior, Eldar can easily get addicted to the pleasures of battle and killing. Many Corsairs eventually grow out of this youthful rebellious phase and return to their Craftworld, though others fall so deep into depravity that their souls are consumed by Slaanesh or they become Dark Eldar themselves. Only those with the strongest wills can stay as a Corsair permanently without falling over the edge. Funnily enough, the Eldar have yet to draw the connection between letting their kids go play pirate and then return home and the Imperium and numerous (usually unnamed) species holding the Eldar responsible for the piracy.
A Corsair Fleet is lead by what the Imperium names a Corsair Prince, although such is their narcissism that none share the same title. These ancient beings tend to be consumed by a singular obsession, be it indulgence in the hedonism that lead to the Fall, quenching their ever-lasting thirst for murder, the pursuit of exotic technologies and elixirs in a quest for immortality, or the pursuit of power through forbidden sorcery. Beneath him are the Barons, each of whom lead a Coterie of Corsair troops. Infighting is very common, with each Corsair harboring within himself the desire to become Prince and each Coterie considering the others within the fleet as rivals.
Corsairs are essentially a pretty irrelevant faction. Despite being one of the Eldar subfactions most likely to meet humans, the fact that they tend to prey on civilian ships means that they rarely get involved in any significant wars unless they're helping out a Craftworld. The only real famous Corsairs are Prince Yriel and Yvraine, both of whom abandoned their roles as Corsair Princes to move onto greater things, and Duke Sliscus, who is just a Dark Eldar who lives on a ship instead of in the Webway. Despite this, Forge World gave them rules as a tabletop faction in their book Doom of Mymerea. These rules had a lot of flavor, and are the main source of lore about their units, but sadly in the shift to 8th Edition they lost a lot of their units, including their commanders, making a pure Corsair army impossible unless you run them as Ynnari. Which you should, since unlike the other Aeldari factions they don't lose any of their gimmicks to do so.
If you want an idea of what life as a Corsair is like, the book "Path of the Outcast" deals with an Eldar from Alaitoc who becomes a Ranger, then joins a Corsair fleet, rises to the rank of Prince, only to make some stupid mistakes and lose it all.
There is a homebrew update of the Doom of Mymerea rules to 8th Ed. at Index - Corsairs.