Alien's score was composed by Jerry Goldsmith, and boy this guy can compose. He's written the soundtracks for movies as varied as Chinatown, Gladiator, Total Recall, Logan's Run, The Secrets of Nimh and a bunch of the Star Trek films. And that's just a sampling. Ever heard of The Ballad of Cable Hogue? No? Well, Goldsmith composed the music for it, too (source).
Goldsmith's style has proven as distinct as the types of movies listed above. About the only through line in his career is that he writes excellent music to heighten the experience of watching movies.
Let's consider some examples of how the music heightens the experience in Alien.
In the opening scene, the camera crawls through the Nostromo, the score underling both the mystery and the grandeur. The mechanical drones of the sleeping ship join the percussion instruments rumbling in the soundtrack, creating a sense of foreboding, while the flutes stay invitingly inquisitive. By the time the crew awakens from hypersleep, the sinister percussion underling the score fades out, and the wind and string instruments build to something lighter, giving the sense of awakening and rejuvenation.
When the alien attacks Brett, the score has an unearthly undertone with some discordant strings that rises to a quick, shrill, violent finale to match Brett's death. Then there is only silence and metallic clang of the chains as Jonesy watches nearby. Unlike the opening track, the music here builds fear and parallels the sudden violence of the alien's act.
Finally, when Ripley battles with the alien alone in the shuttle, there's no music at all. Instead, the sounds of the mechanisms, the alien's movements, and Ripley's troubled breathing build the suspense for us. The music only picks up after the alien has been expelled from the ship, booming and dramatic until the alien is finally killed when it fades into a triumphant exclamation of Ripley's defeat of the monster.
We should point out that Goldsmith's score was edited and rearranged in post-production. As David Giler notes, "This is always a trick part because no matter how much of it the composer hums for you or plays on the piano, you don't really hear his score until you are on the recording stage. And his score was way too lush, more like his Patton score, which was not really what anybody particularly wanted. We wanted it more haunting and weird and strange" (source).
Goldsmith has stated that he originally wanted the main title to play out "very romantically, very lyrically, and then let the shock comes as the story evolves." But he points out, "It didn't go over too well, and Ridley and I had major disagreements over that. So I subsequently wrote a new main title which was the obvious thing weird and strange" (Ibid)
Other changes included Freud. Goldsmith composed this track for the movie Freud, and it was only mean to be a temp track for the hypersleep awakening scene. But Ridley Scott liked it, and, since they had the rights to it, kept it in the final version (source). The ending theme was also replaced with Howard Hanson's Symphony #2.
As with most parts of the movie business, it can be hard to draw the line between Goldsmith's work and the work of others. If you want to explore Goldsmith's original score, you can: it's on CD, online, and even available through DVD extras.