Drop and cast refer to the height and/or angle of the butt stock on a firearm - usually a shotgun, but not always.
Drop is the amount the butt can be lowered or raised at the rear end (the end that mounts on your shoulder). It is measured using the boreline of the barrel as a reference. "Dropping" the butt will raise the reciever when the gun is mounted to the shoulder. "Raising" the butt will do the opposite.
Cast is refered to in terms of "cast-on" and "cast-off". It is accomplished - usually - by steaming and bending the stock either right or left as needed. Cast-on would bring the reciever toward your body and cast-off, away from your body.
Both of these adjustments are made to custom-fit a firearm to an individual. They are usually done by a custom stock maker at a high price. The shims you talk about do virtually the same thing - but only to a point.
