Aeldari is the new, trademarkable name GW invented for the Eldar when 8th edition 40k released in mid-2017. Drukhari is the same thing for Dark Eldar.
Asuryani refers to Asuryan, the chief deity of the Eldar pantheon. Barring only a few exceptions, the whole pantheon was consumed by Slaanesh, according to Eldar legend. It's the name for the Eldar who live on the Craftworlds, which are planetoid-sized cityships. They follow a very rigid lifestyle designed to focus their minds so they don't fall prey to the behaviors that led to the Fall. They're what you think when you say Eldar, but they're not representative of pre-Fall Eldar civilization and they're not the majority of living Eldar.
Corsairs are generally the outcasts from the Craftworlds, who either aren't willing or able to follow the Craftworld's code of behavior or committed crimes that merit exile. When those exiles band together, you get corsair fleets.
Dark Eldar are the real remnant of the old Eldar empire, and they represent the vast majority of living Eldar. They live in a giant city in the webway (a network of stable subspace tunnels, not the Warp but not realspace) called Commoragh. They're absolutely ****ed up, but don't really give a **** about what goes on in realspace. They leave the webway for slaves and not much else. They don't get along with anyone else, including other Eldar.
Exodites are those Eldar who lived on planets on the very edges of the old Eldar empire, and because of that, those colonies survived the Fall. Asuryani tend to be very, very, very protective of them.
Harlequins are completely separate from the rest of the Eldar. Their primary job is the maintenance and protection of the Black Library, a small Craftworld in the Webway that houses all the Eldar knowledge about Chaos. They're servants of Cegorach, the Laughing God, the only Eldar god to have escaped the birth of Slaanesh (again, according to Eldar legend. Whether there's any truth to it is ambiguous).
Dark Eldar are the biggest assholes in the galaxy, but all Eldar are assholes to some degree. To the Eldar, non-Eldar life is worth slightly more than nothing. They don't care about anyone else except in how they can be used as decoys, shields, or weapons.