The movie opens four million years ago, give or take a few, where a group of early hominids lives on the African veldt. They exist in a simpler, technology-free time where scrounging for food, being hunted by leopards, and fighting over muddy water holes is all in an honest day's work.
One day a mysterious black monolith appears before the hominids—because reasons—and soon afterward they begin to develop tools in the form of bone clubs. The clubs allow the hominids to hunt for meat and fight off rival tribes, giving them a distinct advantage in the survival of the fittest.
Flash forward four million years, again give or take, and we find Dr. Heywood Floyd flying to a space station for a layover on his way to the moon. There, he bumps into a group of Russian scientists who want to pick Floyd's brain about rumors they've heard surrounding an epidemic at the Clavius moon base. Dr. Floyd says he's not at liberty to discuss the matter.
At Clavius base, Floyd thanks the resident scientists for their cooperation in the coverup story, saying it's necessary because the public might not be ready to learn about their discovery. Floyd goes to see the discovery himself, and it turns out to be a black monolith identical to the one we saw in Africa. Well, that the apes saw in Africa. When sunlight touches the monolith, it emits an ear-splitting radio signal.
Eighteen months later, we pick up the story aboard the Discovery One, a spaceship bound for Jupiter. Aboard are Drs. Dave Bowman and Frank Poole as well as three other astronauts in hibernated cryosleep. The ship is run by a HAL 9000 computer that utilizes artificial intelligence; that feature was part of the sales pitch, at least. They call him "HAL." He's got a soothing "voice"—calm and measured—but a bit creepy if you ask us.
One day, HAL detects that the antenna's AE-35 unit is about to go kaput. Bowman performs a spacewalk to retrieve the unit, but once aboard the ship, Poole and Bowman find nothing wrong with it. Worse, mission control reports back that its twin 9000 computer says the unit's just fine and HAL is wrong. Like that kid who argues with the teacher about everything, HAL chalks it up to human error.
Houston, they've got a problem.
Poole and Bowman use an excuse to go into the EVA pod, and they secretly discuss what to do with HAL. They decide to replace the antenna unit, and if it doesn't fail, they'll have to shut down HAL's higher "brain" functions. HAL discovers their plan by reading their lips and does what he has to do to protect the mission from these incompetent humans.
When Poole goes to replace the antenna unit, HAL takes control of the EVA pod and cuts his oxygen line. Poole drifts into the cold reaches of space, and Bowman takes another pod out to retrieve him. While the Bowman's away, the HAL will play. He shuts off the life support systems, killing the hibernating scientists aboard the ship.
Bowman returns with Poole, but HAL refuses to let him back into the ship. Bowman uses the emergency airlock to sneak aboard, but in doing so, he has to let go of Dr. Poole's body. Inside the ship, HAL tries to convince Bowman that everything's okay now, he was just having a bad day, worried about the mission, etc. Bowman, not to be persuaded, shuts down HAL's memory circuits one by one, effectively killing him—turning him back into an ordinary machine. It's a dramatic slow death scene as HAL tries to reason with Bowman. He says, "Stop, Dave. I'm afraid." Maybe we can get Siri to say that.
At the moment of HAL's death, a video message comes on the monitor. On it, Dr. Floyd reveals the true purpose of the mission, previously only known to HAL. He tells the crew about the moon's monolith and the signal it's sending. The signal suggests the existence of extraterrestrial life and it's pointed directly at Jupiter.
In Part Four, Bowman reaches Jupiter and locates a third monolith. The monolith opens a Star Gate and Bowman travels through vast distances of kaleidoscopic space-time. On the other side of the psychedelic trip, Bowman finds himself in a classy, neoclassical hotel room. As he explores the room, Bowman ages considerably until he becomes an old man on his death bed.
At the moment of his death, the monolith appears to him and transforms him into a creature that looks like an angelic fetus. No, seriously, that happens. The newly born Star Child travels through space in a futuristic amniotic bubble without the need of a spaceship and returns to Earth.