I can't discuss Moon without giving away major elements of the plot. Therefore:
Moon is the story of Sam Bell—the sole inhabitant of a moon base that gathers energy to be sent back to Earth. He's on a three-year contract, with two weeks left to go, and the solitude has been getting to him—especially since a damaged communications satellite prevents any real-time communication with his family. He's starting to see things, and it's interfering with his work. While working outside the base, one hallucination causes an accident, and he wakes up in the medical bay with no memory of what happened. He wants to leave the base to get the damaged equipment up and running again, but Gerty, his robotic assistant, won't let him leave. He manages to trick the computer into letting him take a quick moonwalk, and once outside the base he investigates the site of the accident, where he finds... another Sam Bell. There follows some great, tense scenes between the two, as they alternately try to pretend that nothing strange is going on and figure out their bizarre situation.
Before long they are able to squeeze the answer out of Gerty: they're clones of the original Sam Bell, who left the base twelve years ago. Their "three-year contract" is actually a capped life-span, at the end of which their bodies begin to deteriorate and are incinerated. Communications with Earth are being artificially blocked, and the taped conversations he's been having with his wife are fake. Everything he's been living for is an illusion, and, in the Dickian tradition, he's forced to cope with realization of the truth.
Sam Rockwell is a pretty amazing actor, and this movie is a fine showcase for that—it's virtually a one-man show (though I certainly don't want to downplay the contribution of Kevin Spacey, the voice of the very HAL-like Gerty). In the hands of a lesser actor this might have ended up schlocky, but he powerfully communicates the soul-wrecking disillusionment the two Bells experience. (The scene in which the older Bell finally establishes contact with his daughter on Earth is particularly devastating.)
As one might expect from a movie about clones, the core of the story involves questions of identity. Each of these beings truly, completely believes that he is the original, real Sam Bell, that he will return to Earth at the end of his contract, that his wife and infant daughter are waiting for him. And who's to say they're wrong? They have the memories and the emotions that go along with them. They have, dare I say it, the soul of Sam Bell. But the company that runs the moon base treats them as objects, as machines like Gerty. But there are hints that Gerty may have emotions of his own. He does seem to malfunction a bit, like HAL 9000. But rather than going on a homicidal rampage, his malfunction manifests itself as compassion. He's programmed to help Sam Bell, and help him he does—at the film's end, he's instrumental in sending one of the clones back to Earth. There are multiple Turing tests going on in this story, and both clone and machine pass them with flying colors.