How can I get the cup stains out of this table?

If marks remain: petroleum jelly or mayonnaise
Rub a thin layer onto the rings.
Leave for several hours or overnight.
Wipe off and buff.
For stubborn cloudiness
Try a small amount of white toothpaste (not gel) mixed with baking soda.
Rub lightly with a soft cloth following the grain.
Wipe clean immediately and polish.
Finish with furniture polish or wax
This evens out the sheen and protects the surface.