Simply stated, a contact force is a force experienced by a solid object due to contact with another solid object. Pushing and pulling involve contact forces. So does running, or simply standing. Collisions involve large contact forces that operate for a very small duration.
Before we do anything else, it must be explained that the word normal here is meant in the mathematical sense of the word, which is perpendicular rather than the opposite of abnormal.
When two solid objects are brought into contact, each of them exerts a normal force on the other that prevents them from merging together. The magnitude of the normal force depends on the force with which the two objects are pushed together.
An example of ...