Continuity mistake: Just after the scene where R2-D2 is riding shotgun on Luke's X-Wing fighter and is hit by a shot from Darth Vader, there's a shot of C3-PO standing beside Leia in the rebel HQ, and the dent on C3-PO's head is on the right side (it's on the left side throughout the rest of the movie).
Continuity mistake: When Luke and Ben enter the cockpit of the Millennium Falcon during the meteor shower, Ben enters the cockpit twice between shots.
Continuity mistake: When Ben says to Han in the Millennium Falcon, "In my experience there's no such thing as luck", you can see Han is resting his arm on the surface near the control panel, but in the next scene his arm is no longer on the surface.
Continuity mistake: After C-3PO is separated from R2-D2 on Tatooine, when he's walking among the sand dunes, the sky alternates between being clear and cloudy.
Revealing mistake: During Darth Vader and Ben's lightsabre duel, sometimes you can see a white wire attached to Ben's lightsabre, for example after he says, "If you strike me down, I shall become more powerful than you can possibly imagine."
Continuity mistake: When everyone is getting in fighters for the final battle, they all have white helmets with blue insignias. However, they aren't wearing them when they're flying.
Audio problem: [Special Edition only] In the scene where Luke and Leia swings across the missing bridge, a number of the shots fired by Leia are "bangs" instead of the laser "zap" sound. The bang is probably from the prop gun used (it shot blanks) and was never dubbed over (until the 2004 DVD release).
Continuity mistake: Right before Luke yells at Han after he fires his blaster in the trash compactor, there's a wideshot where Han is sitting among some garbage and holding his blaster with one hand. In the next shot, he has turned the other way and is standing, holding the blaster with two hands.
Continuity mistake: When Leia's holo-message is being played in Luke's garage, and C-3PO explains about R2-D2's restraining bolt, the background behind the hologram has been flipped.
Continuity mistake: When the rebel leaders are studying the tactical illustration of the Death Star, it shows a large, round indentation positioned at the Death Stars equator. The Death Star actually has a smaller, round indentation in its northern hemisphere. Can't be due to intentional differences between the plans and the finished thing, otherwise the rebels would have no way of knowing that the vulnerability was still there either.
Continuity mistake: When Han yells at Chewbacca to "Get in there, you big furry ape", Chewie steps in. It cuts away, then cuts back and Chewie steps in a second time.
Continuity mistake: In the final battle, the leader of the fighters (not the group that was to blow up the Death Star, but the leader of Luke's group), got killed. Then it goes back to the home base where Leia, C3PO, etc are watching the action and you hear a transmission from that leader.
Continuity mistake: In the scene where the Millennium Falcon is travelling through the meteor shower, Chewbacca roars and you can see he has nothing on his head, but in the next shot he is wearing a pair of headphones.
Continuity mistake: When Ben and Luke enter the cockpit of the Millennium Falcon after leaving Tatooine, Ben is seen standing behind Chewbacca on his seat, but in the very next shot he is seen standing by the seat.
Continuity mistake: When the walls of the trash compactor start moving, Han is holding the pole, and leans it against the wall to his left (we hear it hit the wall). In the next shot, the pole leaning on the wall to his right instead., and the right side of the pole moves up and down between the next few shots.
Revealing mistake: When Luke is first practicing with his lightsaber on the Millennium Falcon against the remote, the zapping blasts that it sends at him are in 99% of the case headed nowhere near his body, but disappear in line with it [very poor FX work] and he flinches.