I made this picture for you - http: //i41.tinypic.com/9r3vb5.jpg (There's space between : and //, so delete it to see the pic)
So, we have a big circle and a small one that is inside of the big one. How do I delete everything on the small circle except the part that is in the big circle?
There might be a simpler way, but the only way I can think of right now is using the Pathfinder effect...
Make a copy of the two shapes.
Group the objects
In the top menu, go to Effect > Pathfinder > Intersect
That will give you the part of the smaller circle that you want...and you can just place it in position on the original larger circle.
Using a clipping mask will keep your original layer shapes intact. The Pathfinder tool will combine/cut/delete sections which is useful if you need to create a new shape (say you wanted to use the "football" shape where your two circles overlap, and add a stroke around it).
Usually my bottom layer is the shape I want for my outline, so I copy it and place it above my other layers. Then, with all relevant layers selected - the colored box all the way to the right of each layer on the layers palette will be highlighted - I create a clipping mask (Object > Clipping Mask > Make).