Dark Aeons are optional bosses that surpass the main antagonist of the game's storyline. Defeating each of them is mandatory for the superboss Penance to appear.

Do you get anything for beating dark aeons?

Defeating each of the Dark Aeons will unlock Penance, the strongest superboss in the game.

Why do dark aeons exist?

When it comes to the dark aeons, the reason for their existence is pretty clear, to get revenge on Yuna and her guardians for destroying Yunalesca, and as far as the believers knew, all hope of ever being rid of Sin.