Fur worn over the shoulders and hanging down?

I believe you are asking about a garment called a 'stole', probably most often heard as a 'mink stole'. This is a garment narrower than a cape, worn about the shoulders, with ends often left dangling. I have seen mink stoles and fox stoles. I believe they became popular in the late forties-early fifties when fur coats were very desirable, and those who could not pay a thousand dollars for a coat might wish to buy a short jacket or a stole to enjoy the prestige of owning mink or other fur.