Continuity mistake: Just after Obi Wan kills the two aliens in the Cantina, there are two quick "reaction shots" from both ends of the bar. If you are quick, you can see that they are both shots of the same end of the bar with one shot flipped. You can see the same human with the light blue/dark blue suit on in both shots.
Continuity mistake: After Darth Vader kills Obi-Van Kenobi, Luke shouts "No!" and then there is a gunfight with stormtroopers. During this, R2-D2 is seen going up the ramp into the Millennium Falcon. But a few shots later, when Han Solo says "Come on!", you can see R2-D2 at the bottom of the ramp and he going back into the Falcon for a second time.
Continuity mistake: In the trash compactor, Han says, "One thing's for sure, we're all going to be a lot thinner." In the next shot, Han and Leia are both standing in a totally different position. For example, both of them are looking in another direction, and Han's left hand is in a different position. (01:20:15)
Continuity mistake: During the conference in the Death Star, the back of Tarkin's chair is higher in the wideshots than in the close-ups.
Continuity mistake: When the walls of the trash compactor start moving, the bar Leia uses to brace them is easily obtainable, but in the previous shots, it was partly buried in garbage.
Continuity mistake: If you watch Leia while they're in the garbage compactor, her belt, at various points, has fallen down around her thigh. It moves from her thigh to her waist (depending on the camera angle) during the whole scene.
Continuity mistake: When the Walrusman attacks Luke in the cantina, if you look carefully you can see that the alien has curved hooves instead of hands (behind the scenes photos confirm this.) When Ben Kenobi cuts off his arm, an arm with hairy paws is lying on the floor. We know it is walrusman's arm because the other guy wasn't wearing an orange jacket.