An executive director is a member of the board of directors and an employee of the company. He/she will have a specific role such as finance director and as such be responsible for the day to day running of the company within his sphere.
A non executive director is a member of the board who is not an employee but appointed for his/her expertise and takes a part in decision making at board meetings. Sometimes non executive directors may get involved in such things as disciplinary hearings, especially for high ranking employees.