Water rings and spots are caused by moisture trapped in the finish when wet objects are left on the furniture. They appear as white cloudy areas. There are several ways to approach the problem:
A substance with oil in it, furniture polish for example, can be left on the spot overnight. Wipe off the oil in the morning. With luck, the spot will be gone.
Alcohol can be used to remove the spot. Dampen a soft rag lightly with alcohol and wipe the area gently. Be careful. Alcohol can damage some finishes, especially shellac. Use a small amount of alcohol and proceed cautiously.
Mild abrasion can remove the water damage. Use rottenstone and/or pumice with a rubbing oil (available at finer hardware stores or Constantine's Woodworkers Catalog 800-223-8087). Proceed with care. You don't want to rub through the finish. #0000 steel wool (available at hardware stores) will also work, but it is more abrasive. Rubbing out the water mark may alter the sheen of the finish. If this happens, you will have to rub out the entire surface to get an even sheen.
Another solution is to coat the entire surface with the proper solvent. This will work with shellac or lacquer. If the finish is shellac use alcohol. Use lacquer thinner on lacquer finishes. You will need spray equipment for this method. Spray the entire surface with the proper solvent and the allow to dry. The solvent will dissolve the finish and allow the moisture to escape. When it dries, hopefully the damage will be gone.
If none of these things work, you may have to strip and refinish the furniture.