As far as I can see, there cannot be such an animal.
The terminology used throughout company law is that directors are always appointed, or they can't be one. So it seems nonsensical for an animal called a non-appointed director to exist.
Well, it depends on the company structure. In some large organisations, the term "director" "VP" or GM could be used as a form of seniority within certain senior managers and in this situation, the said Director is not appointed by and accountable to the board of directors. Hope this helps?