A Kabal is a sort of clan or House of Dark Eldar, with a big heap of organized crime family (and, obviously 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, Reavers, 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 they offer different boni depending which one you choose since the 8e codex dropped
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
- Kabal of the Wraithkind
- Kabal of the Slashed Eye
- Kabal of the Black Myriad
- Kabal of the Flayed Skull
- Kabal of the Lords of the Iron Thorn
- Kabal of the Obsidian Rose (Sounds like some Sisters, tbh.)
- Kabal of the Baleful Gaze
- Kabal of the Bloodied Claw (Definitely sounds like a possible Blood Angels successor chapter.)
- Kabal of the Poisoned Tongue
- Kabal of the Fiend Ascendant
- Kabal of the Dying Sun
- Kabal of the Broken Sigil (Sounds like a CSM warband to me...)
- Kabal of the Crimson Woe
- Kabal of the Ebon Law
- Kabal of the Last Hatred (Edgy!)
- Kabal of the Onyx Scar (That'd be the exact opposite of a White Scar, wouldn't it?)