Continuity mistake: When Luke and Obi-Wan enter Docking Bay 94 in Mos Eisley, as the first shot ends, C-3PO is turned to his left and is standing right behind Obi-Wan on Obi-Wan's left. In the next shot, Threepio is turned to his right, and is standing further away from Obi-Wan, behind him on Obi-Wan's right. (00:52:25)
Continuity mistake: In the scene where Alderaan is destroyed, the "buns" on each side of Leia's head are positioned higher up than in the previous scene she was, so that her earlobes are visible. When Luke and Han rescue her, her hair is back to normal. She's held prisoner in a cell, so it's not likely that she could or would have changed her hairstyle.
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.
Continuity mistake: In the scene where Vader chokes the imperial officer with the force, watch him before he gets choked. On his left (your right) of his blue and red button things, he has a pocket with what looks like two pens sticking out. But in a later shot, only one "pen" is sticking out. Later, it goes back to two again.
Continuity mistake: After the droids land in the desert C3-PO has a streak of oil running down his left shoulder. A couple of scenes later when he is walking after he splits up from R2-D2 there is a long shot of him and he still has the streak on his left shoulder. There is an immediate close up and the streak switches to the right shoulder.
Continuity mistake: When Luke and his uncle buy C3-P0 and a white-and-red R5-D4 robot from the Jawas, the R5 unit blows up after travelling a few feet. There's a quick shot of R2-D2 back at the Jawa's transport, and behind R2 you can see the same white-and-red R5-D4 unit being worked on by the Jawas. In the next shot it is back over with Luke. (00:19:15)
Continuity mistake: When the rebel leaders are studying the tactical illustration of the Death Star, the superlaser's dish is positioned on the battle station's equator. The Death Star's dish is actually offset from its equator, in the station's northern hemisphere. This was because the early CGI used to create the tactical readout of the plans took so long to make that by the time it was finished, the Death Star's design had been changed to offset the superlaser dish, and due to 1970s technology, there was no time left before the movie's release to change the plans, which were accurate to an earlier design of the station.
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). (01:56:20)
Continuity mistake: The microphone on Red Leader's helmet switches from one side of his face to the other, then back again as he pulls out of the Death Star trench (just before he crashes). (Actually the entire shot is mirrored. Check out the scuff marks on the centre of his helmet, they change direction as well.) Other shots and pilots' mics switch too: Red Leader when he says, "All wings, report in", Wedge after a shot of Leia and C-3PO monitoring the battle, Luke when Red Leader says, "There's a lot of fire coming from the right side of that deflection tower" and when Luke says, "I'm hit, but not bad".
Continuity mistake: When Luke, Han and Chewbacca attack the prison of the Death Star, in one shot the end of the corridor is in a slope while it is supposed to be horizontal. (It looks to me like it's actually a flat panel painted to look like the corridor. The angle looks perfect for the opening shot of them entering the prison floor, it just looks "off" for any other angle.)