How to delete certain part of a shape in illustrator

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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).