Shredded condoms were used to create tendons of the beast's ferocious jaws.
The chestbursting scene was NOT filmed in one take (despite the myth). The scene was filmed twice: on the first take, the chestburster was unable to cut through Kane's shirt, so the crew needed to reset and shoot it again. The failed attempt is visible in the finished film, since director Ridley Scott thought it made it look like the creature was struggling to push its way out, and made the scene more violent (see The Beast Within documentary where this is discussed). According to designer Ron Cobb, the two takes of the scene were shot from multiple angles, and about 40 minutes of footage was available for screening.
According to Yaphet Kotto, Sir Ridley Scott told him to annoy Sigourney Weaver off-camera, so that there would be genuine tension between their characters. Kotto regretted this, because he really liked Weaver.
The blue laser lights that were used in the alien ship's egg chamber were borrowed from The Who. The band was testing out the lasers for their stage show in the soundstage next door.
For the chestburster sequence, Sir John Hurt stuck his head, shoulders and arms through a hole in the mess table, linking up with a mechanical torso that was packed with compressed air (to create the forceful exit of the alien) and lots of animal guts (which, according to Sigourney Weaver, caused the set to smell horribly). The rest of the cast were not told that real blood and guts were being used, so as to provoke genuine reactions of shock and disgust. Apparently, this worked so well that Yaphet Kotto went home in complete shock afterwards, locking himself in a room and refusing to talk to his wife for several hours.