What's the difference between a stepping ring and an adapter ring?

I picked up a treasure chest (box of random stuff) full of vintage photography filters and such.

In the box are stepping rings and adaptor rings.  I know what stepping rings do (they let you attack a different size filter to the lens) but what's an adapter ring for?  

The ones I've seen are adapting one thread size to another. For different brand filter systems, gadgets, or whole lenses. There are specialty setups where you attach one type of lens backwards in front of another or with a bellows. There are all kinds of fun oddball arrangements.