I found this comprehensive article in Dr. Stephen M. Pribut's Sport Page
It's a great guide to selecting a running shoe.
The function of a running shoe is to protect the foot from the stresses of running, while permitting athlete to achieve his maximum potential. While in some parts of the world athletes run and participate in sports barefooted, most of us require and benefit from the use of sport specific shoes. The forces and motions that occur in different sports vary greatly. Because of these differences it is important that active participation in varied sports will require varied shoes. A simple example of why this is so can be demonstrated by a brief contrast of the sports of running and tennis. Tennis and other racquet sports require much side-to-side motion and the shoe must provide lateral stability. The shoes appropriate for racquet sports usually do not have any heel elevation. If the shoe is unstable when the athlete is moving to one side to return a ball, the likelihood is great that they may suffer an ankle sprain. Recreational running on the other hand usually occurs in a straight line. Lateral stability is not as important. These shoes usually have slight heel elevation that will reduce stress on the Achilles tendon, but slightly reduce the lateral stability of the ankle. Running shoes also have a larger toe box, more shock absorption, and better pronation control than tennis shoes. Interestingly enough, many of the so-called walking shoes have characteristics that are more similar to tennis shoes than to running shoes. Walking and running both occur in a straight line and the similar requirements of these activities suggest that one would be better off using running shoes for walking, rather than a shoe that resembles a tennis shoe, unless, of course, you walk down the street practicing your backhand returns.
Long distance runners usually contact the ground on their heels. Sprinters have forefoot contact. Middle distance runners vary and may have forefoot, midfoot or heel contact.