How do I easily remove a black ring from finished wood?

Have you tried bleaching the area?

http://www.instructables.com/id/Removing-Black-Stains-in-Wood-Furniture-With-Oxali/

What material is the table itself made of? I would use something that would remove a metal stain and then tend to what I would need to then repair the table from that. Otherwise I might do a slight sand if a wood table.

Oxalic acid is the active ingredient in Barkeeper's Friend cleaning powder. I use this stuff all the time for lots of applications from minor stain removal to polishing glass.

I just applied a paste of Bar Keepers Friend to a dark ring on a wooden desk and wiped off after several minutes with a damp cloth, the ring is gone.

Hometalk recently discussed this exact question. Try doing a search to see if you can find the article. It offered various ways of removing rings on dining tables, occasional tables, etc.

Try rubbing a little toothpaste on the ring to see if it comes off.

http://www.instructables.com/id/Removing-Black-Stains-in-Wood-Furniture-With-Oxali/

Trial and error! Furniture polish and clean cotton cloth. You may have to do in stages