The only entirely foolproof way I know to remove deep, long-set water marks is to sand to bare wood, restain (if your piece was stained) and refinish. However, fresh, less deep stains may be more easily remedied, as I've found with some quick internet research.
Doing a quick google search has found the following home remedies. I have not tested any of these, so take it as hearsay. What a lot of them have in common is oil and abrasion, or a combination thereof, so there may be something there.
- Make a paste of 1 tablespoon baking soda and 1 teaspoon water. Gently rub the spot in a circular motion until it disappears. Remember not to use too much water. (Reader's Digest)
- Apply petroleum jelly and let it sit overnight. In the morning, wipe the watermark away with the jelly. (Reader's Digest, Dummies.com)
- Mix 1 teaspoon salt with a few drops of water to form a paste. Gently rub the paste onto the ring with a soft cloth or sponge and work it over the spot until it’s gone. Restore the luster of your wood with furniture polish. (Reader's Digest)
- Gently rub some non-gel toothpaste on the wood with a soft cloth. Then wipe it off with a damp cloth and let it dry before applying furniture polish. For even stronger cleaning power, mix equal parts white toothpaste and baking soda. Rub the paste parallel to the wood grain, wipe it off, and then polish with lemon oil. (Reader's Digest, Dummies.com, Bob Vila, Charles & Hudson)
- Mix equal parts vinegar and olive oil and apply it with a soft cloth while moving with the wood grain. Use another clean, soft cloth to shine it up. (The OP has indicated that this solution worked) (Reader's Digest)
- Try using a dab of car wax. Trace the ring with your finger to apply the wax. Let it dry and buff with a soft cloth. (Reader's Digest)
- Empty your clothes iron of all the water inside, then bring it into the room with the affected piece of furniture. Plug in the iron, then lay a cotton napkin, towel, or T-shirt over the stain. With the iron set on low, apply it briefly to the fabric before lifting the cloth to see if the ring has diminished. Repeat until (we hope) the white stain has disappeared. Alternatively, you can try using a hair dryer to achieve the same effect. Move the dryer back and forth over the area for about 10 minutes until the moisture evaporates. (Bob Vila)
Most sites I have read suggest applying a furniture polish or wax after the water stain is removed, which I think is a good idea.
This is why we keep a ton of coasters around our house, and also why I yell at people to use one when they have a drink :-)