How centripetal force applies to other common arrangements of a center and an object moving in a circle around it.

Well,

Centripetal force is a special force applied perpendicular to the velocity of the object.  Centripetal force, if applied long enough, will result in circular motion.

I suppose an example would be something like this:
You are holding a milk gallon full of milk.  You begin spinning with the milk gallon, allowing your arms to be "pulled out" by the milk gallon.  You are exerting centripetal force on the milk gallon.

If you tell me what you need specifically, I can edit this.