A casing refers to the metalletic material which houses the powder, the primer, and the bullet. Usually this term is found when describing the outer components of either a handgun or rifle round. Shotgun rounds are not referred to as casings, but rather shells.
Any bullet which can be fired from a gun needs to be placed inside a loaded casing in order to store the powder and create enough pressure for the bullet to leave to gun at a high rate of speed. Below is a picture of a case and its other components:
The casing is also what is left in the action of a firearm after the bullet is fired. Casings can either be ejected in a semi-automatic fashion through the recoil of the gun, or casings can be ejected using a manual action such as a bolt or lever.
Other terms: cartridge, brass