Hitbox

A term originating from Fighting Games, which refers to the box-like or circle-like descriptors of where an attack will hit. When a hitbox touches a hurtbox, the result is an attack successfully landing. Hitboxes are typically hidden in-game and oftentimes do not line up 1:1 with the corresponding attack's animation.

Person 1: My sword went through his body, how did he not get hit?
Person 2: His crouch low-profiled the hitbox of your attack.

Hurtbox

A Fighting Game term which refers to the box-like or circle-like descriptors attached to a character's model showing where they can and cannot be hit. When the hitbox of an attack meets a character's hurtbox, the result is a successful hit. Hurtboxes are typically invisible during normal gameplay and do not line up exactly with a character's model.

In Super Smash Bros. Melee, Mr. Game & Watch's shield doesn't cover his entire hurtbox, meaning he can get hit even while shielding.