Any force which is the result of physical contact, as opposed to electromagnetic or gravitational effects. In quantum mechanics, most contact forces turn out to be short ranged electromagnetic forces, making the contact force seem to be a relic of Classical Mechanics. However, there is a sort of quantum equivalent: degeneracy forces.
Degeneracy forces do not have a simple power law for arbitrary wavefunctions, but for the degenerate gas it follows an inverse thirteenth power law. This makes it quite local when compared to the inverse square law of gravity or an electric monopole, or the inverse cube law of a dipole.