Contact forces are forces that require the actual contact (touching) of two pieces of matter. There are a variety of contact forces. A very common one is friction. Anytime that two surfaces are in contact with one another, there is friction between the two surfaces. The surfaces don't have to be solid either. A skydiver still experiences friction as he "slides" through the air. Specifically that is fluid friction. Other types of contact forces are elastic, spring, and tension forces.
A field force is a force that works at a distance. No touching is required. Gravity is a good example of a field force, because it works whether or not an object is touching something or touching nothing at all. Two other forces of this type are electrical force and magnetic force.