Continuity mistake: When Dallas is in the air vent trying to flush the Alien out of the ship he is carring a flamethrower. Right before his demise, he is told by Lambert to "Get out of there" or something to that effect. In the next shot you see him descending a ladder clearly WITHOUT the flamethrower. However, in the shot after that (when he is lowering himself to the bottom of the ladder, right before the alien gets him) he somehow has the flamethrower again (it even got reslung around his shoulder). (01:15:25)
Continuity mistake: Near the start, when Dallas is at the entrance to Mother, one of the lights to his left, the fifth one down from the ceiling, is not lit. The next shot from inside Mother looking out shows that light is now lit. This exact same thing happens a second time when Ripley is entering mother later in the movie. These two scenes must have been shot at the same time. (00:08:20 - 01:18:35)
Continuity mistake: When Ripley runs back into the control room to abort the self-destruct, one shot shows the timer reading 5 minutes. The shot then cuts to Ripley pulling a lever back to where it originally was and yells out "Mother, I've turned the cooling unit back on! Mother!" The shot then cuts back to the timer where it can be seen it has only gone down by 2-3 seconds, but all of what just happened took longer than that. (01:38:40)
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: Near the beginning when the Nostromo crew are debating over landing on the planet and Parker is arguing against it. Ash addresses him by revealing the company clause that states any potential alien intelligence must be investigated. He even mouths to Parker "no money." In the best shot Ash is directly behind Parker and not in a position where he could have been directly addressing Parker as the previous editing suggested. (00:12:35)
Character mistake: When the crew come upon the space traveler in the derelict space craft one of them says he is fossilized. For fossilization to occur the tissue must be replaced by minerals from surrounding materials. That is why fossils are found in the ground. When the creature dies it must be covered with dirt or mud before it decomposes. Over time the minerals in the dirt replace the tissue. But the space traveler is sitting in the pilot chair, completely open to the environment and not covered with anything. The term they should have been looking for was 'mummification' or perhaps 'skeletonization'.