Catalysts are required for spellcasting, with Staffs used for Sorceries, Talismans for Miracles, and the Pyromancy Flame for Pyromancies.
Catalysts give a Spell Buff stat, boosting your spell damage or healing output. Different catalysts offer different types of attribute scaling, and different base levels of Spell Buff (there are a few rare weapons that act as Catalysts, see the melee weapons for more info on those).