Continuity mistake: When R2D2 is captured by the Jawas, a Jawa welds a restraining bolt to the middle of a panel on R2. When the shot changes to a close up, it's being attached to the top of the panel. The shot changes again, and it's welded to the middle again.
Other mistake: When Han Solo, Luke, Obi Wan, C3PO and R2-D2 are on their way to the Millennium Falcon for the first time there is a scene where they have to walk down a few stairs. If you look carefully you can see that R2-D2 is struggling to get down the stairs but in the next shot he's down with the others.
Continuity mistake: When Luke sees the holograph of Princess Leia for the first time, and says, "Who is she, she's beautiful," Mark Hamill's hair is rather tousled. In the next shot of him, it's neatly combed. Then, when he gets up and says, "Well I don't know anyone named Obi-Wan...," his hair is tousled again!
Continuity mistake: When Luke has been knocked unconscious by the Sand People and Obi-Wan appears and scares them off, there's a shot of Luke lying there and one of his sleeves is in disarray, halfway up his arm. But when Obi-Wan checks his pulse, he does it right through a fully-covered-by-the-sleeve wrist. (00:28:25)