Read up on the major creative movements in film and the techniques which the great directors employed in their work. While a formal education in film theory isn't strictly necessary to analyze film techniques, a little studying can give you a good handle on the basics. Study the basic building blocks of Hollywood films--such as cross-cutting and establishing shots--and then look at the way alternative movements such as Soviet formalism or the French New Wave defied them. Check the works of major directors like Orson Welles, Alfred Hitchcock and Stanley Kubrick, and learn how their unique filmmaking techniques were realized onscreen (see Resources).
Watch films with an open and objective mind. The biggest trick to analyzing film techniques is breaking the habit of passively absorbing a movie and applying a more critical thought process to it. In simplest terms, it means asking why: why we're being shown the specific images onscreen, what information they convey, what kind of visuals are designed to deliver that information and the underlying themes beneath it all. That starts by looking both at the individual shots and the structure as a film as a whole, observing the details closely and considering how they contribute to the film's overall meaning.
Examine the composition of the film's shots and the way they are put together. The formal term is "mise en scene," and includes the way the sets are decorated, the choice of lighting, the movement of the actors and the position of the camera. None of those elements occurs by accident: every one of them is a careful and deliberate choice. Ask yourself what kind of effect they create when placed together and how that enhances either the story or the message of the film.
Look at the way the film is edited together. Individual shots are placed in a particular sequence in order to give a film its structure. They also play a key part in lending the film meaning; for example, a shot revealing who the killer is may be much less effective at the beginning of a scene than at the end. Formalistic film theorists also believe that the organization of shots carries meaning in and of itself--that the very act of linking one shot to another conveys a message separate and distinct from either of them individually. Understanding how editing affects meaning is vital to understanding film techniques.
Look at other films by the same director to see if you can spot recurring themes. The "auteur" theory, advanced by the French New Wave, states that directors (and, by extension, actors, screenwriters, cinematographers and so on) use recurring themes and visuals in their films which leave a distinctive stamp the same way an author does in his books. By watching multiple films from the same director--especially films with wildly different themes or subject matter--you can pick up on those elements and gain greater insight into the creative vision he hopes to convey.