Hardening off plants is the process, usually undertaken in spring in the temperate zone, of preparing plants started indoors for the change in environmental conditions they will encounter when permanently moved outdoors. Without allowing plants to harden off, they will be negatively impacted by the sudden shock of exposure to daytime's sunrays and nighttime's coolness.
The process of hardening off plants involves a transitional period in which plants are left outside during daylight hours only and in an area where they can be shaded and protected from wind. Having a cold frame facilitates hardening off, but it is not essential that you have a cold frame. Watering is reduced as well during the hardening off period. Gradually, the plant is allowed exposure to an increasing amount of sunlight and allowed to stay out later and later.
Occasionally, you'll also hear the word used in reference to winterizing trees and shrubs in the fall; used in this sense, you "harden off" a plant by reducing watering in early fall, as a means of making the plant tougher in winter.