The scope of the director role depends a lot on the specific type of director. For example, a film director doesn't just yell “action” they are also responsible for the film crew, managing scripts, and overseeing editing. Essentially, they manage the overall direction of the film as well as the film production.2
A director of an organization such as a company or non-profit is responsible for setting the strategy of the organization. As part of their role, they will help set goals and budgets and then track progress. Unlike management, they take a “big picture” approach to setting the direction and goals of the organization. Management is responsible for implementing the strategy set by the directors.
Organizations don’t just have board directors. In some organizations, the manager of a specific department may also be called a director. Common examples include the marketing director, finance director, or operations director. There can also be an art director, and director of nursing. Their role is to direct and manage the operations of the department they oversee. Film sets will also have different directors for different parts of the production. These involve the art director, director of photography, and assistant director.
There are lots of organizations that have directors. These include public organizations such as hospitals, universities, and within government departments. Private organizations such as companies and non-profits if large enough will have a board of directors. Directors can be appointed from within the organization or from outside of it. Many companies have independent directors on their boards which helps to bring in fresh thinking and a new perspective.