Pay attention to camera angles. As a movie director, camera angles act as the eyes for the audience. Pay attention in the movies you watch how directors tell the story with their choice of camera angles.
Listen to the way dialog is presented. Directors, as well as a few other people, decide how a line is ultimately said. Of course the actor may at times have more say than a director; it is a principle job of the director to direct the actors.
Look for film techniques such as variation in colors, the grain of the film (meaning how clear the picture looks) or the transitions. All play a part in telling the story and the director chose each one for their contribution to telling the story. Try to see how their decision improves or takes away from the movie.
Ask what other people thought of the movie. Directors are always looking for feedback and based on what people say they can change their style on the next production they work on. If you ask other people what they thought of a movie, you too could improve your viewing skills and see the movie through another pair of eyes.