Silver may turn your finger green if the silver reacts to acids in the skin or other substances on the skin. This problem arises mostly with inexpensive jewelry, or silver-plated jewelry, as opposed to pure sterling.
There are certain chemicals in human skin that react strangely with certain metals, and silver is one of those metals. It doesn't necessarily have to be a sterling silver ring in order to have this reaction, as silver is found in many different alloys used in jewelry.
The reason one gets the green skin is from the chemicals in the skin reacting with the silver, causing it to oxidize and change colors. This also can happen due to lotions, ointments or anything else used on the skin.
One fast fix to help avoid oxidation is to coat the ring in a thin layer of clear nail polish or clear varnish. This wards off oxidation, at least for a while, according to About.com Chemistry.