Revealing mistake: When Qui-Gon, Jar-Jar, and Padme are eating at Anakin's house, Anakin's mum pours water for Qui-Gon. But when she gets to Jar-Jar, she tips the pitcher but no water comes out.
Continuity mistake: Obi-Wan's braided hair keeps changing sides, and getting longer and shorter, most noticeably in the scenes around the underwater city.
Continuity mistake: When Anakin attaches 3PO's right eye, one shot has the eye in his left hand, then the angle changes and the eye is being applied with the right hand.
Continuity mistake: When Palpatine and Queen Amidala are conversing in his office (when Palpatine refers to Valorum having 'very little real power') his outfit switches from a green breasted cloak with a blue collar area to a green breasted cloak and dark green collar. It switches back after one shot.
If you watch closely during the scene in the senate, right after Queen Amidala calls for a vote of no confidence in the chancellor, the camera starts showing different views. The shot after a Senator asks for order is a group of pods with aliens in them. If you look in the lower left, you can see a pod full of E.T's. There's also a pod full of Wookies seen at one point (with their traditional noises, and for the first time in 20 years, the species was played by someone other than Peter Mayhew).
Continuity mistake: When the queen is being escorted by some droids after she has been captured, the shadows change completely. When we first see everyone walking past, their shadows stretch ahead of them, but when we see them walking towards Obi-Wan and Qui-Gon, the shadows are really elongated and at an angle.