Intelligence governs the power of Sorceries. It also controls weapon attack power for weapons that scale with Intelligence. This includes Weapon upgraded with Magic Upgrade and Enchanted Upgrade paths and unique weapons, specifically Moonlight Greatsword and Velka's Rapier.
Sorceries require a minimum level of intelligence to cast.
(You can acquire without the required stats and even attune the spell, but attempting to cast will leave your character scratching their head, literally.)
Intelligence does increases the damage of punch attacks made with The Pyromancy Flame.
NOTE: Some weapons (e.g. Velka's Rapier) require a minimum intelligence level to be wielded properly.
If you do not meet the minimum intelligence required, you will stagger every time you hit an opponent.
List of weapons and their Scaling
Also see weapon scaling.
Scaling - D
Scaling - E
Scaling - A