Hardening: The process of placing steel into it's hardest possible state. Usually measured via the Rockwell scale in knifemaking. Hardness is achieved by bringing the given steel to it's "critical Temprature" and then rapidly cooling, depending on the type of steel, this could be air, oil, or water. Several things occur to the steel during this process, including matrix structure transformations,(this means things happen to the grain in the steel) and the equal distribution of the elements contained in the steel (commonly referred to as "solution")
Tempering: Simply stated, is a controlled softening of the steel. In it's just hardened state, the steel is of course hard, but along with that comes brittlness, which is undesirable in a typical knife blade. Tempering is a way to soften the blade to a "usable hardness" level.
Tempering is accomplished by reheating the blades (after hardening) to a much lower temp, that imparts the desired level of hardness AND durability.
Most everyone has seen a piece of steel that has been ground on a bench grinder or the likes, that has changed colors (blues, purples and such) at the area being ground. What is really happening is that the steel is being tempered (softened). This doesn't mean much.....unless that steel/tool is something used to cut. Many a good tool has been destroyed by someone with a grinder, not knowing what they are doing!