The Ability Sphere allows you to activate Skill, Special, and Magic nodes on the Sphere Grid. Basically, if it grants you an ability, you will be able to activate it with this sphere. There is a lack of these towards the beginning of the game, but Wakka can access an ability in his area of the Sphere Grid, called Extract Ability. Using it on an enemy that doesn’t normally drop Ability Spheres will force them to drop them.
Normal enemies that drop Ability Spheres: