Continuity mistake: There are many holes in the 'squad structure' once the marines arrive at Hadley's Hope. Gorman calls up 1st Squad (comprised of Apone, Vasquez, Crowe and Hudson) to approach the colony first. They do. But, if you look closely, there are other marines present. When they arrive at the door Apone goes straight to the console, Hudson holds back to our left and Crowe rests in the background on the right of the screen. Apone then calls Hudson up to do a bypass. If you look closely as Hudson runs up to the console you will see Frost in the background, he steps up to the left of Crowe (his left) and on the left of the screen, you will see Wierzbowski crouch with his flamethrower, slightly obscured by Vasquez (look between her legs). Then 2nd squad is called up. It is comprised of Hicks, Dietrich, Drake, Frost and Wierzbowski. Watch the shot of Vasquez entering the lock and you will see Frost standing roughly where he was when the 1st Squad ran up. Clearly the original scene had all the marines descend on the North Lock simultaneously. The confusion continues when Gorman, Ripley, Burke and Bishop enter the South Lock. They meet up with Hicks' squad yet Dietrich and Wierzbowski are curiously absent.
Revealing mistake: In the Special Edition of the film, in the scene where the Marines are conducting their initial search of the colony, the shot where Hudson and Vasquez enter the "negative situation" room (where they find the hamster) is obviously sped up. You can tell by how quickly Vasquez seems to move when she comes through the door.
Continuity mistake: When Ripley and Hicks are discussing their defence plans after the first encounter, they decide to put one of the auto guns in each of three locations, leaving one left over. When we see the guns in place they have actually used two guns in each of two locations, that is, two pairs.
Revealing mistake: Right before Ripley discovers all of the eggs that the Alien Queen has laid, look at Newt's fingers as Ripley is carrying her - It's obviously a fake hand. Newt is in fact a dummy in this scene, and others preceding the chase from the Alien Queen. This dummy was created so that Weaver would not get exhausted carrying a real child around for so many scenes/takes.
Visible crew/equipment: In the Reactor Room scene, where the Marines first encounter the aliens, there's a part where Dietrich is grabbed from behind and accidentally ignites Frost with her flamethrower, causing him to fall down the stairwell. As he's falling, if you look, you can see part of a leg and a cable belonging to one of the technicians on the set.