Much of what we know about the Void is actually from VERY recent revelations in the Main Scenario, so a lot of the stuff being discussed here could be considered spoiler territory. You have been warned, OP! (Albeit, perhaps, too late.) Anyway, here's some more spoilers!

That said, it's pretty much been established that the Voidsent that plague Eorzea originated come from the Thirteenth Reflection (the Reflections being alternate universe copies of our world, the Source, that were split off sometime in the distant past for unknown reasons, and which the Ascians want to Rejoin with our world, also for unknown reasons, but presumably because doing so will revive their god, Zodiark). The Void actually is the result of a massive gaffe on the part of the Ascian in charge of the Thirteenth Reflection, Igeyorhm, who pushed the Heroes of Light of that world too hard, resulting in an overwhelming victory by the forces of Darkness. The resulting Flood of Darkness transformed that world into the Void, a dark dimension whose wraithlike beings hunger for the Aether our world can provide and so break through to this world as often as they can. This is NOT something the Ascians want, as the Void cannot be rejoined to the Source - it is a world lost to them, and of no use to Zodiark.

It is unknown what would happen if another world fell to Darkness. Would it become a second Void? Or would it merge with the first Void to become a single, larger Void? It is also unknown how this Void relates to the Void in previous Final Fantasy games; the franchise as a whole implies that the Cloud of Darkness (who pretty much personifies the Void) across multiple Final Fantasy games are the same entity, and that the Void represents the space that separates the different realities that the different Final Fantasy games exist within - but whether any of that is relevant to the Thirteenth Reflection Void in this game, I don't know.