A corn is an area of localised thickening of the skin. This thickening is referred to as 'hyperkeratosis' and occurs as a result of excessive compressive pressure and shearing forces, it is body's way of protecting the deeper structures from damage. Pain occurs due to the pressure placed on underlying nerves.
A popular misconception is that corns have roots; this is false, a corn does not have a root. Instead it has what we call a nucleus. The nucleus is a conical shaped area of hard keratin which has formed in response to pressure and/or friction.