Water marks on tabletops caused by sweaty glasses or sitting water can usually be removed, though not without some risk of damaging the finish.
Professionals have access to an aerosol product called “blush remover” that restores the color to lacquer finishes. Blush remover doesn’t work well on shellac, water base or catalyzed finishes. If you use a blush remover, be sure to spray just a fine mist on the water ring. Don’t make the surface wet, and for sure, don’t touch the surface until it dries, which can take awhile.
Other than using a blush remover, you can try coating the mark overnight with an oily substance such as mayonnaise or cooking oil. Sometimes, this will cause the mark to fade somewhat, but usually not much.
The method that works on all finishes, except high-performance finishes such as conversion varnish, is to abrade off the watermark. The whiteness is almost always right at the surface of the finish, so it doesn’t take much rubbing. The trick is to avoid changing the sheen (gloss or satin) of the finish more than necessary. So your choice of abrasive is important.
Every situation is different. Here are some suggestions for abrasives to try, from finest (and least dulling) to coarsest: toothpaste and your finger; a very fine, usually white, nylon abrasive pad; and #0000 steel wool. You can reduce the size of the scratches (and therefore the dulling) of the nylon pad and steel wool by using a wax or oil lubricant.
If you can’t get the sheen of the rubbed area to blend in with the surrounding surface, you may have to rub out the entire tabletop.
Be aware that heat marks also show up white but are usually more difficult to remove (though the methods are the same). The damage may go deeper into the finish.
Black rings are in the wood and can’t be removed without removing the finish first.