Continuity mistake: Before Lambert is killed, the alien twists its tendrils around her leg, and we see that she has sneakers on. However, when Ripley notices Lambert's dead body (hanging?) she is barefoot. [The rest of the scene was cut. According to the novelization, Lambert was actually intended to be shown completely naked and sexually ravaged. Director Ridley Scott had toyed with the idea of creating the illusion of the Alien having raped her before her death. He thought better of it and cut the scene. Still a mistake, but there's why.] (01:33:40 - 01:34:25)
Continuity mistake: After the alien leaves Kane's face and the rest of the crew are looking for it in the infirmary, there is one shot that bears re-watching. When Ripley is looking upwards over the prone Kane, you begin to see the dangling tentacles of the alien in the background. At the same time, the collar on her outfit begins to pull downward, un-naturally, until the alien drops onto her shoulder. I believe there was some sort of string or rope fed from below, through her clothing, and was attached to the alien and was pulled to have the creature drop onto her. (00:48:33)
Plot hole: The shuttle "won't take four" crew members. Considering it only has two cryotubes (and hypersleep appears to be necessary for any hope of survival), it doesn't seem to accommodate three people, either. (Admittedly I would still go for it, too, if I were up against a 7-foot-tall acid-blooded alien monster with two sets of jaws, but there doesn't seem to be much difference whether there's three or four people onboard).
Continuity mistake: In the escape pod Ripley presses buttons to release gases onto the alien. Watch her helmet between the cuts. During the first two gas releases the inside of her helmet is foggy and streaked; cut to the alien reacting to the third gas and when the camera returns to Ripley the inside of her helmet is clear.
Factual error: In the opening scene we see a bobbing bird on the table. These birds require a wet beak to work thus the glass of water. We find out that the crew is half way home when they are awakened. Later we find that they are 10 months from home, so they must have been travelling for 10 months already, meaning the bird has been bobbing for all that time and the water has somehow not evaporated.
Other mistake: In the opening bridge sequence when the Nostromo's computer wakes up, text is displayed on the monitor. The scene cuts to a shot of the reflection of the text in a helmet's visor. The reflected text should be reversed, but it is not. (00:02:00)
Continuity mistake: Just after Dallas is taken by the Alien in the air shaft, the crew are round a table discussing what they should do next. Ripley has a visible injury to her right cheek, a round swollen injury. In the next scene the injury has gone. It's possible that it was caused by accident filming the scene when she was attacked by Ash with the rolled up magazine, although that happened later in the film.
Character mistake: While entering the atmosphere of LV-426 when the ships starts to shake in a turbulence, there is a close-up at Ripley's screen showing the terrain and at the bottom of that screen we can see "WEYLAN YUTANI NOSTROMO 180246." The correct name of the Company is WEYLAND YUTANI and Nostromo's registration number is 180924609. (00:16:35)
Revealing mistake: As Ripley, Lambert and Parker leave the room after decapitating Ash, Parker turns around and incinerates Ash's body. As soon as the shot starts, if you look carefully just before the flame actually touches Ash, you can see Ash's head (mostly the skin below his neck) has already melted before the flame, or even the heat for that matter, touches him. The flamethrower had only been set off a split second before the fire reached Ash and by that point the head has already began melting, so the heat made from the fire could not have melted his head that quickly.
Continuity mistake: After Ash is decapitated, Ripley says they are going to blow up the ship. In the shot where Parker looks back, Ash's head is slanted to the side. The shot cuts back to Parker where he turns on the flamethrower and incinerates Ash, whose head is suddenly flat on the table. (01:25:40)