Well, far as I can figure... aether is like a combination of matter and energy. Therefore it behaves like both and neither.
Remember, energy cannot be destroyed - it simply changes form from usable to unusable. Sunlight helps grow trees, which can be burned for energy, leaving only unusable ashes. In the world of XIV, this is simply not true - aether changes from one element to the next, spinning on the elemental wheel observed as a part of the UI in Eureka.
That's not to say aether can't become unusable - corrupted aether and crystals are highly volatile and can't be used for anything but disrupting other "normal" types of aether, such as when it was used to help breach Garuda's barrier and endure Leviathan's Tidal Wave on the high seas. If corrupted aether then transitions into a usable state... well, that I'm not sure of.
Either way, one of the major reasons primals are so dangerous is that they introduce true entropy to Hydaelyn - the aether that makes up their physical form returns to the land when felled, but any other aether a primal consumes while on the material plane is simply gone or otherwise converted at a very poor rate into loose aether upon their defeat. (Case and point: Bismarck, known to eat islands in the Sea of Clouds, leaves nothing behind upon death but the key to Azys Lla. Why wasn't that destroyed? Plot, I'd wager.)
It's also scientifically provable that aether levels across Hydaelyn have dropped through the ages, evidenced by Matoya's Crystal Eye (an ancient Crystal of Light) and the Crystals carried by the contemporary Warrior(s) of Light. This is implied to be due to Calamities and primals; where the aether is going isn't explained, but can be presumed to be going toward awakening Zodiark.
As Moose-senpai said, a Flood of Light or Darkness can also destroy aether - Dark compresses it into a singularity, while Light bleeds it away into nothingness.
That's all the information I have on aether, with a little bit of extrapolation by YT thrown in.