From Diablo Wiki
The Black Abyss is where all demons are spawned from.
Being a place that spawns demons, and where demons return when killed. It's likely located somewhere in the Burning Hells, but not necessarily so, since it could, in theory also be the spawning place of angels. In either way, it's separate from the Void, which is more like a "limbo" between planes and places of existence, where beings can be trapped. Lilith is an example of a demon trapped in the Void.
It's primarily Tyrael that brings information about the Black Abyss: