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Xenomorph - Used once, among many other words, to describe the aliens in the franchise, this word stuck as the standard term used by fans. Other names for the species used in spinoff media are 'Internecivus Raptus' (Murderous Thief) and 'Linguafoeda Acheronsis' (Foul-tongue from Hell, Acheron being also the name of the planet on which the creatures were first encountered in 'Alien').
None of the life cycle stages (i.e. Facehugger, Chestburster, Drone) were ever officially named. They were given Fan Nicknames which simply stuck.
The Great Politics Mess-Up: As everybody knows, Weyland-Yutani is a combination of motoring conglomerate British Leyland and generic Japanese. Ah, British Leyland, that pride of the nation, a household name for decades and trailblazer for the world, such an unstoppable industrial force would surely spread its Mega Corp. tentacles across the galaxy for sci-fi centuries to come. Thing is, this film was released in 1979 and British Leyland went bankrupt in 1975. For Britons, the fall of a once proud company was the ultimate symbol of Britain's postwar decline - at least, if you're old enough to have heard of the company in the first place. For Americans, two words - General Motors. Oh, and Japan tanked in the '90s too. But the Asian half of the equation is fine. In 2009, the shattered remnants of British Leyland went bankrupt one last time - and were bought out by the Chinese.
Image Source: This franchise provides the page image for (those for the other films are listed on their respective Trivia pages):
Keep Circulating the Tapes: The "Aliens" Mod for Quake was killed by Fox lawyers (creating the term "Foxed" for that situation), but of course that didn't stop it from being available online for many years afterwards.
Deleted Scene: From near the end of the film, as Ripley comes across Dallas while setting up the ship's destruction, going through the process of being turned into another egg and finishing the explanation of the alien's life cycle. Ridley Scott loved the idea, but found that the scene was too much of a speed bump in the middle of the climax.
The cast knew the general details of what they were shooting for the Chest Burster scene, but they didn't know they were going to get sprayed with real animal blood and entrails. Veronica Cartwright's shocked scream is entirely real.
Cartwright really slapped Sigourney Weaver. That wasn't just a sound effect, and Weaver's recoil and look of shock is genuine. According to the actress in the DVD commentary, she was fed up with Sigourney, who at that point had acted only on the stage and so was not used to pretending to get hit, instinctively flinching away from the slap and so, after numerous failed takes, faked the first slap and then properly hit her when she flinched.
Director Ridley Scott placed a veiled cage with a German Shepherd in front of Jones the Cat, and unveiled it when he shouted "Action!!" Hence when The Alien rose up behind Brett like a phallic gargoyle, the menacing hissing of fear from the poor kitty cat was real.
Yaphet Kotto did a lot of improv acting. Scott played along with it, and advised him to antagonize Sigourney Weaver, so their conflict later in the film would be more believable. When Ripley yells at Parker to "SHUT UP!" after Dallas's death, Weaver already had to listen to Kotto talking over her dialogue dozens of times.
Executive Meddling: According to the audio commentary, Ash as a character did not exist in the original concept at all, and was added during production. This apparently annoyed one of the writers, thinking it an unnecessary and a distracting diversion from the main plot, though the other writer and Scott thought it a good twist.
Heck, this film is one of the few good examples of this trope paying off. The above detail of Ash was introduced by two of the producers associated with Fox, David Giler and Walter Hill, who also extensively rewrote the script to give a more natural flow to characters' lines and interactions. Definitely appears to have paid off.
Fan Nickname: Space Jockey - the alien pilot aboard the derelict ship — extended to the rest of his race, as well. Derived from a name used by the film crew; in the canon, it's never named. In Prometheus they're called the "Engineers".
The adult creature is sometimes dubbed "Kane's Son".
One-Book Author: An acting variant. Bolaji Badejo, the guy in the Alien suit, apparently vanished into thin air after completing the 1979 film. Although considering he was found in a pub and only hired because he was massive and very thin, this is understandable.
The Other Marty: Actor Jon Finch had been cast in the role of Kane, but had fallen very ill due to his diabetes and was replaced by John Hurt.
Take That: One common story is that "Weyland" and "Yutani" were the names of two of Ridley Scott's neighbors whom he didn't like. However, this isn't the case. Ron Cobb, the designers of the movie came up with the name "Weylan-Yutani":
...Weylan Yutani for instance is almost a joke, but not quite. I wanted to imply that poor old England is back on its feet and has united with the Japanese, who have taken over the building of spaceships the same way they have now with cars and supertankers. In coming up with a strange company name I thought of British Leyland and Toyota, but we couldn't use "Leyland-Toyota" in the film. Changing one letter gave me "Weylan," and "Yutani" was a Japanese neighbor of mine.
Working Title: The first film was known as "Star Beast" in its earliest stages. When the writer went through the script he saw characters constantly referring to the Alien, and then the title came out at him, noting that is both a noun and an adjective.
Ridley Scott and some of the cast mention in the audio commentary that he had the idea that casual sex between any and all members of the group regardless of sex, and that, in hindsight, he would have liked to show a homosexual relationship. A scene related to this would have been a conversation between Ripley and Lambert, one clarifying that they had either had sex or been solicited for sex by every man on the ship except Ash, as a way of foreshadowing that he is, in fact, an android.
The original script had a Downer Ending where the alien kills Ripley, then imitates her voice in a call to Earth, indicating it's headed there next. Scott was persuaded that the film was horrific enough without it.