Abilities were first only for Cocoon citizens who didn't have any talent in magic.
They gained non-magical abillities that anybody could posses like Attack
which is a really simple ability.
Then there where some advanced abilities like Blitz or Jump
that also is non-magical. That was the only thing the Coocons could use in battle.
But then the L'cies came and gained abilities who where struck my magic that grew even stronger
than the Cocoons abilities. They gained abilities like Focus And Meditation.
So Basicly Abilities are small techniques that is unique to the user and can give
them Advantage sometimes.