Black Mage was kind of answered in Dalex's post, as well. Shatotto eventually discovered how to channel her own aether, along with that of her surroundings, to power her spells. The lore on the Black Mages is tied to Mhach which, only having done BLM up to 50 so far, I can't really say much.

I've always thought the aether in XIV close to the Mist in XII: it's everywhere, you absorb it just by walking and any magic effect is just careful manipulation of it. Even the Espers in XII and the egis in XIV are kind of similar; in XII, you deplete your mist reserves (MP in XII) to summon an Esper after you defeat it, while you only use up a relatively small amount in XIV to summon your egi (lore-wise, it's your aether shaped into a form similar to said primal).