Continuity mistake: In the shot of the alien's tail wrapping around the back of Lambert's legs right before it kills her, the floor is different to what it was before, there is water suddenly falling from the ceiling when there wasn't before and her trousers have changed from white to blue. This is because that shot was originally intended to be used when Brett was killed.
Revealing mistake: Mirrors were used to make the corridors appear longer, early in the film when Brett is complaining to Parker about the way they are treated by the company you can see their reflections in the mirror behind them as they walk down the corridor. It's only for a second as they are moving off camera but you can see them in their white overalls.
Continuity mistake: During the alien chestburster/death of Kane sequence, Kane's shirt switches back and forth from being tucked-in, to being un-tucked throughout the scene. (00:56:15)
Continuity mistake: When the search party goes into the boneship, they are wearing full space suits, and hoods under their helmets. When Kane's helmet is first cut off, he has no hood on. (00:36:45)
Deliberate mistake: When the Alien appears behind Brett, we first see its tail lowering before the rest of its body, even though the tail is presumably how the creature was being suspended in the air. Immediately after chomping Brett's head, he and the creature are lifted upwards, again presumably by its tail, though it's not long enough to carry them the implied distance and it's unclear what it has latched onto in the first place. This scene starts a tradition of sorts, as both 'Aliens' and 'Alien 3' feature their own scenes in which an Alien grabs a victim and both are subsequently lifted upwards by a great height, presumably by the creature's tail, and the mechanics by which this is possible and even where exactly they're going off to are always obscured to the point where the creature may as well be wearing a jet pack.
Factual error: When the crew shoots Kane's body out into space, the door opens up and "explosive decompression" causes Kane's body to fly out into space. At the beginning, the body slowly rotates. Then as it moves further away, the body begins flipping much faster. The reason something would increase speed in rotation like that is due to aerodynamics. An object traveling through space would simply keep the same rotational velocity it begins with since there is no air or other influences (gravity, etc.). The explosive decompression may cause it to increase rotation speed, but by the time it begins to flip, any air would have dissipated into space and not work as a column of air/wind to force the body into a flip.
Visible crew/equipment: After Ripley fails to shut off the auto destruct sequence, she runs through a corridor filled with gas. As she exits from the gas right before the shot cuts away we see her pass by a full sized light. Beside this light is the clear arm and head of a lighting crew member. (This is acknowledged on the DVD commentary).
Revealing mistake: In a scene near the end, the camera operator appears to bump into Sigourney Weaver from behind, prompting her to momentarily break character. Weaver had been frantically running down a corridor with the camera following closely behind. When she reaches a ladder, the view passes her for a split-second and bounces back a step from an apparent soft collision. Weaver glances back at the camera with a look as if meaning to say, "Are you all right?" but immediately resumes climbing the ladder. (01:39:35)
Continuity mistake: In regards to the self destruct mechanisim of the Nostromo, It is said that the time from the setting to detonation will be 10 minutes. This is also confirmed by the voice of "Mother" who says, right after Ripley initiates the sequence, that the detonation will take place in "T-Minus 10 minutes" However, in the next shot you see one of the monitors which shows a countdown with 10 minutes and an extra 30+ seconds on it. (01:35:55)