According to Woody Allen there are two kinds of directors: “the ones who have it, and the ones who don’t.”
It is well-known that there are two more kinds of movie directors: the ones who write their own material like Tarantino, and the ones who adapt, like Alfred Hitchcock, or Steven Spielberg.
From a more technical perspective, there are two other kinds of directors; the ones who arrive to the set knowing exactly what they are going to shoot and how (the Coen Brothers), and others who arrive with just an idea and follow their gut (Woody Allen). When it comes to directing styles, there are also two approaches; some directors watch the actors rehearse and then decide with their DP where the camera should be, and how many shots will be needed. Some do the opposite. They walk around the set, decide where the camera goes, where the action will take place, and then ask the actors to accommodate.
We could probably add two other kinds of directors: the ones who make films because they have something to say, and the ones who make movies to find what they have to say.