It depends on what color the rings or stains are and the surface film, if any, on the woodwork.
White rings (stains) are indicative of moisture entrapped in a film finish such as shellac or lacquer. You want to displace or release the moisture to clear up the finish. Here's a schedule of events from the mildest to the most severe, so try them in the listed order.
1) Apply mineral oil or petroleum jelly to the area and let stand overnight. Repeat applications of this poultice if it shows any beneficial effect until all clear.
2) Denatured alcohol, applied sparingly with a swab or tampon (a rolled up piece of gauze). You MUST test the alcohol for adverse effects first on an inconspicuous area before use, as it will dissolve a shellac finish. You should see results in about one minute after application to the ring.
3) Rubbing out, the most destructive and last choice, can be done with 0000 steel wool and mineral oil as a lubricant. Stroke the ring with the wool, always WITH the grain, trying not to cut all the way through (removing) the finish. This will remove the sheen from the area, and it will have to be evened out with pumice powder, and finally rottenstone, again using mineral oil as the lubricant.
Mineral oil can be cleaned off with naphtha, but again, test all chemicals on an inconspicuous area before use.
Note these methods will not work for black rings, they are a totally different type of damage and beyond home remedies; scraping, sanding and refinishing will be necessary.
Hank Metz, A Woodworker's Notebook http://www.ameritech.net/users/hankm/index.htm