Abilities are skills that strengthen and support a character.
They are held by equipment (weapons・armor・accessories), and you can set them by equipping said item or learning that Ability.

You can set Abilities by using Ability Points.
By unequipping Abilities, you will recover Ability Points.
(Ability Points will increase in accordance to a character's level.)

By equipping an item that holds an Ability, you will be able to set that Ability.
When you win a battle, the items you have equipped will gain experience, and if the conditions are fulfilled, you can learn the Abilities they hold.
You can then set the Ability without having to equip the item.