A cinematographic technique which states that the camera must remain on the same side of an imaginary line, perpendicular to the camera's viewpoint, from which the establishing shot is taken. The 180 degree rule is an important element of the continuity style.
Imagine two people standing face to face. Draw a line from the centre of the top of Person A's head to the centre of the top of Person B's. Now extend this line to infinity on both the x and y axes, dividing the two people bilaterally. To follow the 180 degree rule, the camera must, in each sequence of shots, stay on one side of this line. If this rule is not followed, the characters will not appear to be addressing each other and the cinematic illusion will be broken, as the viewer's sense of perspective is disrupted.
The 180 degree rule also applies to other shots: for example, in a chase scene, if a car exits the frame to the left, it must enter the next shot from the right. Sports broadcasting also follows the 180 degree rule, as the spectator would quickly become disoriented if the camera moved from one side of the field to the other.1
This rule, like any rule of the cinema, can be broken. Ingmar Bergman crosses the 180 degree line in Persona, while many non-Western filmmakers (who have, perhaps, not been so firmly indoctrinated with the rule's sanctity) ignore it. Finally, it is usually agreed that this rule may be disregarded during a tracking shot. Of course, when the camera stops moving, a new 180 degree line is created in accordance with the new POV.
Belgand pointed out to me that The Two Towers contains a clever violation of the 180 degree rule: in the scenes in which Gollum/Smeagol converses with himself, the camera moves back and forth across the imaginary line. The effect mimics that of a shot reverse shot sequence, which is conventionally used to portray a conversation between two people. Here, of course, the two people exist within the same body, hence the disorientation. In other words, Jackson (or his cinematographer) shoots Gollum/Smeagol as if they really were different two people, not simply two aspects of the same personality.
1Sports video games usually do not follow this rule, which has confused me on occasion (I'm looking at you, FIFA 2001).