There are two types of water damage to furniture: black water marks and white water marks.
A black water mark means that the moisture has gone through the finish and into the wood. The black color is the result of the moisture reacting with the tannic acid in the wood. As a homeowner, there is nothing you can do to remove the black ring without ruining the furniture. A professional can usually get the black out but it involves refinishing the affected surface. I recommended that you leave the black water mark and restore the existing finish. The black mark imparts character to the heirloom piece. It does not detract from its historical, sentimental, or economic value.
White water marks are caused by moisture penetrating the protective finish (Shellac, varnish, lacquer, water based finishes, catalyzed lacquer, conversion varnish, spirit varnish, urethane, etc.) The moisture is trapped below the finish but has not penetrated the wood. It is very important to treat the white rings immediately. White water marks may turn to black water marks depending on the integrity of the finish. The older the finish, the more likely the white water marks because the finish is breaking down. White water marks are caused by moisture from glasses, hot pizza boxes, and casserole dishes, etc.
Immediate action: The sooner you take action, the better your prospects of success. The moisture must be drawn out of the finish. If you do not see cracks in the finish, apply petroleum jelly to the affected area. Let it stand over night. Paste wax is my second choice. This can be applied to any finish.
If that does not work, call me. The next progression of remedies depends on the type of finish. Different finishes require different procedures.