In Final Fantasy III, there are three categories of spells; Summons, Black Magic and White Magic. You’ll be able to recognise many of the spells used in this game from other Final Fantasy games, as well as the Summons themselves, including Shiva, Ramuh and Ifrit.
Below you’ll find the list of Class-types which can use those said spells.

Black Wiz – Black Magic
Conjurer – Summons
Hunter – White Magic
Red Wiz – Black & White Magic
Sage – Black & White Magic, Summons
Shaman – White Magic
Summoner – Summons
Warlock – Black Magic
White Wiz – White Magic

You can also find more information about the different spells by viewing the sub-categories of this page.