Continuity mistake: In the wide screen version, after Luke, Han, Chewy, Obi-Wan and the droids come out of the hidden cargo bins in the Falcon, Obi-Wan says "Who's the more foolish, the fool or the fool that follows him." Then he tries to lift himself out of the bin...it looks as if he doesn't quite have the strength to do it - he puts his arms on the side, exerts a bit of pressure, and waits for the cut.
Revealing mistake: In the scene at the end of the very first battle, where Imperial troops have just taken control of Princess Leia's rebel ship, Darth Vader breaks the neck of the rebel leader and tosses his body to the wall. The "dead" rebel puts up both of his hands to avoid slamming his face into the wall.
Continuity mistake: In the scene where Luke and Leia are about to swing across the canyon inside the Death Star, Leia shoots one last time. She hits the bulkhead behind the stormtrooper on the right and he falls forward. The trooper on the left continues firing (as seen in the screencap).
Visible crew/equipment: Just after the Falcon takes off from Mos Eisley, there's a shot of the Falcon with Star Destroyers chasing. The next shot is of Han Solo alone in the cockpit. Watch carefully behind him in the doorway, and you'll see the shoulder of a man in a green shirt quickly move out of shot.
Revealing mistake: Just before Lord Vader appears the first time (on the boarded consular ship, through the breached door), there is a Stormtrooper checking another trooper apparently dead on the floor (he's lifting the dead one's head). Then Vader enters, and the Stormtrooper quickly straightens up, dropping the head. It turns out the "dead" Stormtrooper was actually alive, however, because he carefully lowers his own head to the floor.