A Kabal is a sort of clan or House of Dark Eldar, with a big heap of organized crime family (and BDSM) thrown in.
There used to be proper royal houses in Commorragh, dating back to the Eldar Empire, and persisting even after the Fall. They operated cautiously, raiding Imperial worlds infrequently, so as to avoid attracting the ire of the Imperium. Asdrubael Vect didn't much like their aristocratic/nepotistic ways (having been a slave before clawing his way into a position of relative power), so he arranged their downfall by intentionally raiding the Imperium, eventually luring the Salamanders into Commorragh to wreck things. Since he was expecting them, his Kabal of the Black Heart stayed intact, and cut-throat (figuratively and literally), meritocratic Kabals like his became the ruling bodies of the Dark City.
Each Kabal is headed by an Archon. Unlike most mob bosses, Archons are often in the trenches, getting into the thickest part of the raids, as hanging back is a sign of weakness, and in Drucharii society weakness is death, or rapid replacement, most likely by way of death. Archons are also almost always the best fighters in their Kabal (see below for why). Under them are the Dracon, the officers of the Kabal and leaders of the deadliest squads, who often plot against their Archon. Then come Kabalite Warriors, who are themselves segregated into the "Trueborn", those born from an actual mother and father (and potentially more actors are involved in the process if you believe some of the older lore), and the mass-produced vat-grown "half-born". At the very bottom are the slaves who do all the menial work and, when they aren't good for anything else, get tortured to feed the souls of the Kabalites with their pain.
Kabals often work in parallel with other Dark Eldar organizations. Most Archons sign a contract with a coven of Haemonculi, promising new victims and working material (often the two are the same) in return for access to resurrection chambers and the assistance of whatever horrors the coven has at its disposal. Wych cults, Hellion gangs and other unsavories looking for glory may tag along (or be paid to come along) on a raid.
A Dark Eldar who stands out from the crowd can earn prestige and leadership, but the ambitious can also move up in life by killing their immediate superiors and taking their place. This is not seen as problematic by Commorragh society -- in their eyes, a leader who cannot stop his subordinates from killing him obviously isn't much of a leader. On the other hand, he's also considered not much of a leader if no-one wants to kill him.
In game terms, The Kabals offer different bonuses depending which one you choose since the 8e codex dropped, though as of right now only a select few have out and out rules, that's never stopped you from plucking some off the generic list and plunking thematically appropriate ones onto Your Dudes, did it?
There are a shit-ton more than these, especially petty ones operating out of Low Commorragh, who might as well be pissant Necromunda gangs. The fun comes in though when you hear a name that could, perhaps plus or minus a little grammatical change, just as well be that of an edgy Space Marine chapter as a Deldar Kabal or Coven (read the Path of... series and you'll encounter a bunch not enumerated here which to be perfectly honest I could not tell you if cross-quizzed which category they were in...White Flames, anyone?)
- Kabal of the Black Heart - the most powerful Kabal in Commorragh, headed by Asdrubael Vect.
- Kabal of the Poisoned Tongue - The "subtlest" Kabal, while also one of the most openly dickish. Headed by Lady Malys.
- Kabal of the Obsidian Rose - The arms dealers of Commoragh, run by a former sweatshop slave who went absolutely bugfuck nuts and now holds a stranglehold on the weapons trade in the dark city.
- Kabal of the Dying Sun - a bunch of Boomers who swear up and down they still can blow up suns if they feel like it, and thanks to their absolutely insane archaeotech, they can. But more often than not end up blowing themselves up in the process.
- Kabal of the Last Hatred - A bunch of truly Edgy necromancers who plan to somehow transcend mortality, and kill their craftworld counterparts to a man.
- Kabal of the Flayed Skull - A bunch of truly bloodthirsty murder pirates that zerg rush pretty much everybody in their way on-site. Also one of the fastest growing Kabals, apply today and get your own Jetbike!
- Kabal of the Broken Sigil - (Sounds like a CSM warband to me...) And you'd be right! As they take their entire raiding strategy from the Night Lords.
- Kabal of the Lords of the Iron Thorn - The makers of the skimmers that the Dark Eldar depend on, and brought themselves back up from nothing after a failed rebellion.
- Kabal of the Black Myriad - Nothing known about them but their name and symbol.
- Kabal of the Bladed Lotus - Nothing known about them but their name and symbol.
- Kabal of the Bleaksoul Brethren - Known for a twisted sense of humor and their shark like teeth.
- Kabal of the Wraithkind - A Kabal that got on Vect's bad side, and are now forced to essentially be a bunch of Mandrake vampires who can only feed on pain in the material plane. Understandably, they are nonplussed about this.
- Kabal of the Baleful Gaze - Make use of beast masters and wych cult allies.
- Kabal of the Falling Moon - Rivals of the Dying Sun Kabal. They are just as old but originate from a pre fall lunar cult instead of solar one.
- Kabal of the Slashed Eye: (Now a front for the leader/folk hero of all Hellion gangs, Baron Sathonyx).
- Kabal of the Bloodied Claw (Definitely sounds like a possible Blood Angels successor chapter.)
- Kabal of the Fiend Ascendant
- Kabal of the Crimson Woe
- Kabal of the Ebon Law
- Kabal of the Onyx Scar (That'd be the exact opposite of a White Scar, wouldn't it?)
- Kabal of the Jade Knife