Intelligence is a stat in Dark Souls III that increases attack damage with intelligence based weapons and catalysts as well as increased damage output with spells such as sorceries, pyromancies, and hexes.
Intelligence increases damage done by weapons and catalysts with a scaling mechanic. A letter denoting a weapons specific scaling properties is noted beneath the weapon's required intelligence stat. The letters range in order of effectiveness from E, D, C, B, A, and S.
Weapons and catalysts with an E rating for scaling will receive a poor benefit from investment in intelligence while weapons with an S rating receive the greatest benefit.
Note: the term Catalyst refers to any medium used to cast spells.
As is with every other stat in the game, it also has indirect bonuses such as increasing the player characters flat damage absorption.