Revealing mistake: When Newt is sliding across the grated floor, Bishop reaches out to save her and you can clearly see he's standing in a hole to make him look cut in half. (02:11:40)
Continuity mistake: During the scene where Ripley is trying to break the window with a chair, there is a scuff mark on it before she hits it. It goes away and comes back. (01:32:15)
Suggested correction: The view from Ripley's side is looking up at the window and from the other side level where there are reflections at the same height (using Ripley as a reference) of the scuffs. There is no way to tell if the scuffs are there or not.
The Director's Cut (and maybe even on the theatrical version) on the 2010 Blu-ray removes the scratches, so I assume they were not supposed to be there initially.
Continuity mistake: In the end of the film, when Ripley pulls the Alien queen down in to the hatch we can see that the bulb on top of the machine cracks but when they are lying on the hatch the bulb is whole. (02:05:10)
Continuity mistake: When Bishop is impaled by the Queen alien he spews android fluid/juice all over his chin. In a subsequent shot his chin is clean and he is spewing fluid again. The following shot his chin is once again covered. (02:23:30)
Continuity mistake: At the very end, when Ripley is closing the lid of Hicks's sleeping capsule, we see a shot of him and the bandage almost covers his left eye. In the close-up shot of the lid closing, the bandage is further up on his forehead.
Continuity mistake: Hicks has a heart painted on his armor with a latch locked by a padlock. When Hicks and Ripley enter the elevator after losing Newt, the padlock is missing. After the Alien attack, the padlock reappears.
Continuity mistake: In the shot were Ripley is driving the crew away from the aliens, note the big dent in the vehicle. Then in another shot when they finally get outside, there's a shot from the side and the dent is gone. (01:19:15)
Continuity mistake: When Ripley is giving Newt the cup of hot chocolate the handle is on Newt's right. When it shows the close up shots, the handle is on Newt's left. (00:48:40)
Continuity mistake: After the Alien Queen is sucked out of the airlock, Ripley climbs back onto the platform and closes the doors. In a shot from above, her left leg has not yet crossed the mouth of the opening. Cut to a closer shot and half her body is already on the platform.