A case is the component of a firearm cartridge that contains the projectile (bullet), the propellant, and the primer.
Cases are normally made of brass, although steel is becoming common for low-cost ammunition. Some premium brass cases are plated with nickel. Other exotic materials have been used, including plastic and stainless steel, but they are uncommon.
The word “casing” is incorrect.
A gun case is a container for safely transporting a firearm.