How do magic rings work?

Are you talking about the type that can connect and unconnect?

If so, then one of the methods is by a hinged part of the ring. Also, they sometimes have gaps in them that a good magician can keep hidden.

If I've not understood the question properly, can you clarify?

If you are talking about ornamental rings then:
They don't work.

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If you are talking about "Linking Rings" then that's easy. You get a set of rings, 2 are linked together, three are linked togethe and then there is a "key" ring that is actually split or cut. The cut isn't visible because it's a clean cut and when the parts are together the ring appears to be complete and uncut.

For the most part the "magician" give you the appearance that all the rings are individual rings. There's one complete ring that is handed to the someone in the audience to inspect. The rest of the show is a simple matter of manipulating the rings to make them appear to connect and disconnect. The "key" ring allow you to connect the sets of rings making them look like a chain.