Continuity mistake: On the journey from the Cantina to the Millennium Falcon the camera pans left to right to follow the actors as they walk first towards and then away from the camera, towards the hangar. At the beginning of the shot R2-D2 is heading in one direction, the droid leaves the frame momentarily and at the end of the pan, trundles into the shot from an impossible angle.
Continuity mistake: Several times during the Death Star battle when the Rebel ships are flying down the trench, their position in relation to the chasing TIE Fighters change. For example, sometimes the Imperials are just behind and then nowhere in sight when, next shot, they are back in close firing range.
Other mistake: Widescreen version only: When the stormtroopers are searching for hiding Rebels in the blockade runner, Leia pokes her head out from her hiding place. In the next shot, in the lower left hand corner, something white sticks out behind the foremost "pillar" and then disappears out of view.
Continuity mistake: Near the end of the Vader/Obi-Wan duel, Luke and Co. run up to the Millennium Falcon. Luke says something along the lines of "Ben?" and it cuts to a shot from roughly behind Luke that shows Vader and Obi-Wan. Look closely: Obi-Wan already has his lightsaber straight up and Vader is going in for the kill. It then cuts to a shot of Obi-Wan glancing over at Luke. Obi-Wan then proceeds to raise his lightsaber (again) for the famous Jedi-disappearing trick as Vader slashes at him (again).