White (non-gel) toothpaste is an excellent mild abrasive for removing water rings from the surface of your finished furniture. Simply squirt a small amount of toothpaste onto the water ring. With a clean cotton rag, wipe the toothpaste in circles over the water ring until it begins to disappear. Wipe up the excess toothpaste with another rag.
Mayonnaise contains a small amount of acid that helps counteract the moisture that's left in the water rings. Dab a bit of mayonnaise onto a clean cotton rag and begin rubbing it into the area with the water ring. Cover the area completely and let the mayonnaise sit from a few hours to overnight. Wipe off the mayonnaise with a clean cotton rag until the surface is clean. It's also a common practice to combine ash from a cigarette or pipe to the mayonnaise as a further means of absorbing the moisture from the water ring.
Mix a small amount of baking soda with filtered water into a paste-like consistency. Dip a clean cotton rag into this paste and rub it over the white ring. Rub back and forth in the direction of the wood grain. Repeat this process, dipping into more paste as needed, until the white ring has been removed.
If none of the above methods work, the moisture that caused the white ring is well below the surface of the finish and onto the wood itself. In that case, you will have to strip and refinish the furniture or hire a professional refinishing and restoration service to do it for you.