Despite sharing the name in the English localization of Final Fantasy XIV, the abilities known as Aero III in the title are in fact the recurring Aeroga spell in Japanese
Aero III is a tier III Black Magic spell. An upgraded version of the Aero II spell, it deals Wind-based elemental damage to a single target. It serves as a preferred method of inflicting wind damage to single targets during the level 60 - 80 leveling period for the pure casting jobs that have access to the spell. Black Mages in particular can take advantage of Aero III when paired with Quake's wind-elemental resistance lowering effect.
Aero III deals massive wind elemental damage to the entire party. It is usable by the Malboro Queen in Final Fantasy XI dungeons.