The first thing you'll notice about Alien's ending is that it isn't exactly revolutionary. There's no twist ending, no revelation, no moment that makes you shout at the screen, "What is happening? Has the world gone mad?!"
But does that mean this isn't a great ending to an otherwise classic movie? Not necessarily.
Back to Normal Normal Land
Which, if you're curious, is just like NeverNeverland only you grow up and die. Super depressing, but at least there are no murderous mermaids.
The alien's presence twisted the crew's normal life into a horrifying dark world version of their world. The Nostromo was a place of safety and protection until the alien turned the ship into a dark-Darwinian battle for the survival of the fittest.
Here's a refresher of the ending: After discovering the alien on the ship, Ripley hatches a quickie plan. She slips into a space suit, forces the alien out of its hidey-hole, and then opens the door to blow it into space. The alien uses its prehensile tail to climb into the ship's engine, and Ripley hits the thrusters, killing it for good. We then end on Ripley's final log:
RIPLEY: Final report of the commercial starship Nostromo third officer reporting. The other members of the crew Kane, Lambert, Parker, Brett, Ash, and Captain Dallas are dead. I should reach the frontier in about six weeks. With a little luck, the network will pick me up. This is Ripley, last survivor of the Nostromo, signing off. (Alien)
After killing the alien, Ripley returns to the status-quo, snoozing in the snug little hypersleep pod where she and the crew awoke at the film's beginning. The landscape of the Nostromo, which was breaking down into a place of chaos and confusion, has been replaced with the shuttle with its reassuringly normal, alien-free tech. In the loving embrace of the ship, she sleeps soundly.
Welcome to the Jungle
The alien is the ultimate hunter, nature's survivor in its purest form. Ash says as much when he tells the crew, "You still don't understand what you're dealing with, do you? The perfect organism. Its structural perfection is matched only by its hostility" and calls it a "survivor—unclouded… by conscience… remorse… or delusions of morality" (Alien).
With that in mind, you can read the ending as showing human ingenuity and technology once again triumphing over nature. Space travel is essentially mankind breaking its natural bonds through technology. When the alien arrives, the crew attempts to use technology—from surgical tools to flamethrowers, cattle prods to motion trackers—to subdue it. But the natural force that is the alien just keeps on slaying.
In the finale, Ripley asserts her dominance over the alien by using technology and some quick thinking. The spacesuit and harness she uses to keep herself safe, the grappling hook that knocks the alien out of the ship, and the engine used to cook it Kentucky-fried style—they're all tools.
Of course, there are other ways to interpret the symbol of the alien in the film, and these various interpretations will lead to different readings of the ending. Did you read the ending differently?
A fun fact about Alien is that it's one of the progenitors of the Final Girl horror trope. As Roz Kaveney points out, "In 1979, the slasher film had not evolved to the point where Ripley is instantly recognized as the Final Girl, the androgynous female without vices who will always make it through to the last scene" (source).
In the '80s and '90s, horror films took this model and ran with it. Every horror film from Scream to The Nightmare on Elm Street featured a final girl who survived fisticuffs with the big bad monster/killer. And they were all brunettes, too. Huh.
Thankfully, Ripley managed to avoid another horror favorite, Sudden Sequel Death Syndrome, making Aliens a much more enjoyable film than some other sequels we could mention.