Maybe I should have started here. It seems all roads lead back to the director.
The director is the artist. There are many jobs on a film or television set held by artists - like a Production Designer - but the director is THE artist.
A film is the director's interpretation of the script. They've decided the style and tone and cast the actors. They've chosen the location for the shoot, and most likely will tweak the script as it rolls along.
Everyone on the movie is supporting the director's vision.
Many times the script or the idea comes from the director originally and this is when you will see a director also listed as a producer. Other times a director is sought after by a producer, or team of producers, because they believe that director can either help them achieve their vision, or will have a vision of his or her own that will make an idea come alive. (Certain franchise movies such as Harry Potter come to mind, where each sequel may have a different director.) In those cases, the director can be replaced if his or her artistic vision doesn't mesh with the producers or the studio.
Logistically, the director has their hands in all of the creative decisions from pre-production through to the final locked picture. He or she is the one who has mapped out the scenes to shoot and has decided, along with the director of photography, the camera angles. The director is the one who is coaching the actors on their moment, and is the one who calls "Action" and "Cut." A director is not - in the year 2010 - usually dressed as The Gorilla is here.
A movie is a director's baby. Think of your favorite films and you can usually name their directors: Hitchcock, Spielberg, Allen, Marshall. In the past decade or so it might be Guest or Apatow for a few laughs. But could you name their producers? Their cinematographers? Rarely.
A director has all but signed his name on a film as a painter would sign the corner of the canvas.
I hate to thoroughly confuse you, but there are also multiple types of directors. There is the director of photography and assistant directors, all of whom are very integral to THE director's job. By the time we're done, you'll know the difference between all of them.
The Gorilla is also an Executive Producer and a Producer, and it's fairly common for directors to have both titles. More explanation on those titles - and the differences between them - to come. Eager to learn more? Associate Producers and Production Designers are the other titles I've covered so far in the End Credits Education.