Welcome to Stack Exchange! Moonboots and Robin Betts both have good, robust answers. Here's another one that might be interesting.
Some advantages to this method:
- It can be very fast if the cutout is the same radius as the sphere.
- No change in topology are necessary.
- UVs are not affected.
- It only works on UV spheres that haven't been rotated.
- It's very difficult to make the hole a perfect sphere whose radius of curvature is different from that of the original sphere.
Assuming you're starting with the default UV sphere without rotation:
Select a vertex that will be on the edge. Set the position of the 3D cursor to the position of the vertex (Shift+S -> Cursor to Selected)
Also make sure pivoting is around the 3D cursor
Select all the faces in the inner sphere (Select top vertex, Ctrl+Numpad plus)
Then invert the vertex positions along the Z axis (S, Z, -1, Enter)
If you'd like the radius of the inner sphere to be smaller than that of the outer one:
Add more faces to the inner sphere with loop cuts if necessary (Ctrl+R)
Scale the vertices in the inner sphere along the z axis, so the hole is as deep as you'd like. (S, Z, Drag)
Imagine where the center of the inner sphere is in space. Set the 3D cursor to that position.
Then it's possible to align all the inside vertices to a sphere with the "To Sphere" tool (Alt+Shift+S, drag)