I think centripetal force is whatever is pushing/pulling the object toward the center of rotation, such as the closed door of a car pushing on you while driving around a curve, where otherwise you would fly out of the car. Another example is the wheels of the car causing it to travel on a curve instead of straight. Or the rope of a tetherball for a pulling example.

In most cases (besides orbits in space) the force is question is actually the electromagnetic force, like any other case where objects made of atoms touch.

Personally I think it's weird to call that a specific force, especially by those who don't want to give centrifugal force a name - sure it's really just things "tending" to travel straight instead of following the curve, but no reason that can't have a special name, it's certainly intuitive enough.