I would like to know the cheap but effective way to remove or cover it. It is a white-ish color, I assume the coffee reacting to the clear coat? Am I close? How do I fix it? Thanks, The Comet would eat the table. . . I would not remove the finish, just a little coffee ring.

seriously..try comet. it works for me on my wood tables!
Your water ring soaked into the varnish, shellac or lacquer. There are a some tricks to fixing this but you need to know what type of finish it is. It is too risky to give you advice without knowing. There are some products out there, maybe something Formby’s makes, that are labeled as water ring removers. Chk Home D or elsewhere.
If the ring is light, it may dissipate on its own. Too heavy and it will remain because it soaked into the wood as well. The top will have to be refinished completely. Refinish any furniture that could get a water spill on it with Polyurethane.
I’ve had really good luck with a mixture of mayonnaise and ashes. Apply the mixture, let it set for several hours or overnight, then wipe away. This has worked several times for me. I did a quick google and found the same advice online:
http://www.thriftyfun.com/tf939047.tip.html
I also found some interesting advice on an eBay forum for removing water stains with a household iron. I haven’t tried this myself but the photos posted there are incredibly impressive. You may want to give that a try if the mayo and ashes doesn’t work for you. Here’s the link:
http://forums.ebay.com/db2/topic/Miscellaneous/A-Little-Giftfrom/520141217
Good luck!