Continuity mistake: When you first see the Millennium Falcon, it is in the hangar. If you look close enough, you can see that only half of the ship is there. This is because it was too expensive to build a full-scale model.
Continuity mistake: When Han Solo and Luke Skywalker escort Chewbacca up to the Cell Bay to rescue Leia they are still wearing the stolen Stormtrooper uniforms, including white boots. Yet after they escape the trash compactor they are back in the boots they were wearing when they escaped Tatooine. While we can believe the rest of their outfit can fit under a Stormtrooper uniform, the boots wouldn't.
Revealing mistake: During Kenobi's final lightsaber battle with Vader, Kenobi tips his lightsaber downwards, right before a scene change. The white "glow" that should be there isn't, showing the stick-like prop the actors used in filming. There is a small white light at the tip, too. This was not fixed in the Special Edition, but it was finally fixed on the 2004 DVD. (01:31:25)
Continuity mistake: When Vader picks up the rebel in a choke grip, the legs of the stormtrooper in the doorway behind them are apart. In the shot of the rebel's feet, the trooper's legs are together. When Vader tosses the dead rebel into the wall, the trooper's legs are apart again.
Visible crew/equipment: After the heroes have escaped the trash compactor and we see them arguing outside of it. As they walk away and Han says "No reward is worth this," the shadow of the camera covers his whole face.
Continuity mistake: Special Edition: Luke, accompanied by the droids and Obi Wan, is in the speeder heading into Mos Eisley, but just as the shot ends it looks like they're about to run over a few pedestrians... they're heading full speed at a group of three or four people just when the camera angle changes. (00:40:50)
Deliberate mistake: When Obi Wan deactivates the Death Star's force field, there's an electronic gauge going down and a sign in clear English with the word POWER - as we've seen in shuttles and other places, they don't use English. The studio decided it was better to show "power" and "tractor beam" in English so that the audience would know what Obi-Wan was doing. The lettering we see in Jedi doesn't really affect the plot, so it doesn't need to be English. Still doesn't really fit though. This is fixed on the DVD.
Other mistake: In the beginning of the movie, when Leia is uploading a message into R2, we can see C3PO looking for R2. When they show C3PO, there are rebel soldiers behind him, running through the corridor, being shot at and falling down. If you look closely, you can see a black stripe vertically running down in the middle of the doorway.
Continuity mistake: When Vader tells Tarkin that Obi-Wan is in the Death Star, as Tarkin says, "The Jedi are extinct." he is standing between two chairs. In the next shot, he is standing behind one of the chairs. The distance between some of the chairs around the conference table changes during the scene as well.
Continuity mistake: Special Edition: When Han is talking to Jabba the Hutt, as Jabba says, "Why did you fry poor Greedo?", the man standing to his left is pointing a blaster towards Han. In the next shot, the man is no longer pointing the blaster at him. (00:51:00)
Continuity mistake: When Vader is strangling the Rebel aboard the blockade runner, the Rebel's hands keep changing position between shots, as he tries in vain to loosen Vader's grip.