More or less we have the details straightened out here; but to add another phrasing...
As far as we know:
Hydealyn and Zodiark once shared the aetherial sea. As existence crept along, Light and Dark drifted apart, creating a binary. Zodiark (The Will of the Dark) lusted for more power and in taking it threatened the balance, so we are told. An act of self-preservation, Hydaelyn (The Will of the Light) exiled Him away from the planet and bound Him within the moon.
This act was so extreme that it stressed the fabric of the universe so hard that it cracked. Like a broken mirror, reflections of the original world (new dimensions), or shards, fanned out in all directions, carrying with them varying balances of Light and Dark of their own. Each of these dimensions is now a world of its own. Our fourteen total dimensions started identically but have been developing independently ever since.
They are other dimensions, we just don't know how big they are. Are there really entire universes? No idea. The Source is the original world - the mirror that all the shards were broken from. It is the dimension that we live on, the dimension where Hydaelyn still resides.
This status quo leaves Zodiark powerless and alone; he has been quarantined for his alleged transgressions. The Ascians are loyal to the Will of the Dark and seek to see him resurrected. They see the fractured universe as an aberration of nature - an abomination.
To "heal" the universe, they cause a great deal of chaos on the Source and one of the reflected dimensions to weaken the boundaries between them. If that wall comes down, everything on the reflected universe is annihilated and its aether is returned to the Source in a great deluge of chaos and imbalance (a Calamity). They believe that this aether can be funneled to Zodiark to help his resurrection.
It's implied, but not confirmed, that the Source experiences repeated Calamities, but the reflected worlds (shards) only experience the one that destroys their world and returns all aether to the Source.
The great disasters that the people of the Source remember as "Calamities" are such events. Seven worlds have been rejoined. However, one world has also been completely lost. The Ascians in black don't have any concept of moderation, they play to win. Unfortunately for everyone involved, if either side wins too hard, the world is annihilated by a Flood of Light or Flood of Darkness that completely eradicates all aether and leaves a void in its wake. The Thirteenth was lost in this manner and is the home of all the monstrous entities we know and love.
This is one place Cilia and I have divergent theories, or rather, he jumps farther ahead than I do. Light and Darkness have always acted similarly, from Floods to sprites. The "blank white perfection" (EN only) refers to events that are happening; the world hasn't fallen yet, isn't devoid of aether, the denizens aren't yet starving souls battling over scraps, etc. Urianger says that the denizens of the First, like the denizens of the Thirteenth, would never again know life or death; he makes it sound like Arbert is worried about more than simple extinction. It's possible that light- and dark-born voids behave differently, of course, I'm just playing it safe until further information. With all the similarities on both sides of the binary, perhaps the emptiness left in the wake of a flood might be the same regardless of whether it was Light or Dark's fault.
As far as we know, Elidibus is the arbitrator. He stops the Ascians in black from winning to hard, he stops the Warriors of Light from winning too hard. He keeps wars ever escalating in perfect balance so that the calamities will rejoin the aether to Zodiark and, upon his resurrection, the Source will still be here for Him to claim.
I theorize that if the Ascians were made impotent, and Zodiark rendered silent, the rest of the dimensions could co-exist for all time if Hydaelyn, too, sat out and let Her children thrive. But we'll see. Hints do seem to say that both Light and Dark need to gtfo, but there's more than one way to go about that. We could cause them to re-merge and annihilate each other, as well ... we're just not sure mortal life would fare well in that case, so I'm sticking with the former, for now, lol.