Continuity mistake: In the Detention Block, Han and Luke are in their stormtrooper outfits and are firing at the enemy. They go to a shot of Han then to Luke. Luke's white shoulder pads of his stormtrooper outfit are clearly missing, and you can see a black undergarment. They then go back to Han and then back to Luke again, where the shoulder pads are now back. (01:15:20)
Continuity mistake: In the scene where stormtroopers are firing at the Millennium Falcon as it leaves Mos Eisley Spaceport, watch the trooper on the far left. After he is shot he begins to fall, then suddenly he's standing again in the next shot, where he begins his fall again. This happens two or three more times, this falling and straightening.
Continuity mistake: In the wide screen version, after Luke, Han, Chewy, Obi-Wan and the droids come out of the hidden cargo bins in the Falcon, Obi-Wan says "Who's the more foolish, the fool or the fool that follows him." Then he tries to lift himself out of the bin...it looks as if he doesn't quite have the strength to do it - he puts his arms on the side, exerts a bit of pressure, and waits for the cut.
Continuity mistake: In the scene where R2-D2 is rolling down a path on Tatooine, under the eyes of scavenging Jawas, the path he is traveling on can be seen to end several feet in front of R2, and beyond it is a rough, rocky surface that R2-D2's wheels could not possibly traverse. In the next shot, R2 is rolling down a suddenly extended (or perhaps the same?) path towards capture.