How do I get water rings out of my wooden coffee table?

Unfortunately, once the damage is done, nothing short of refinishing will remove those rings entirely. However, here's a trick you can use to minimize their appearance in the meantime: Sand the rings down a little, then make a paste of cigarette ashes and vegetable oil to rub into the wood. As long as it's not too big of an area, this should nicely darken the wood nicely so the rings don't stand out as much. Good luck!