2001 covers millions of years of plot, from prehistoric times and on into cosmic timeless space future (or what was the future in 1968, when the book was written). Yet, despite that, the plot actually isn't all that complicated.
A big part of that is because there's no real character development to speak of. Another part is that a lot of the book is just description of astronomical niftiness. The result is that 2001 manages to cover a lot of ground without much plot. A lot of whoosh, not a whole lot of bang.
The novel starts millions of years in the past, with a bunch of man-apes. Being a man-ape kind of sucks; you only eat berries, you're always hungry, you die young. But then a space slab falls to earth and starts tinkering with the man-apes' minds. Moon-Watcher, our hero man-ape, gets especially smart, and figures out how to use tools. He kills a pig for food, then a leopard, then the head of a rival tribe. Man-apes are moving on up.
Millions of years pass, till 2001 or thereabouts (a year that was thirty years in the future when Clarke wrote the novel). Dr. Heywood Floyd (who has less personality than the space slab) goes to the moon to look at a big slab the moon colonists have excavated called TMA-1. Just as he gets there, the sun's rays hit the slab, and it starts broadcasting a signal. (Good timing, Dr. Floyd.)
Cut to a few months down the space road. David Bowman and Frank Poole are in the ship Discovery, originally headed for Jupiter, but now diverted to Saturn at the last minute. They're going to the Saturn moon Japaetus, which is where the TMA-1 broadcast signal was directed. Bowman and Poole don't know about the aliens for reasons that don't make a lot of sense (other people on Earth definitely know), but then again, we've also got giant slabs on the moon, so what's a plothole or two between friends?
Anyway, Bowman and Poole weren't told about the aliens; only their computer Hal was told (and three crewmembers in suspended animation). This was supposed to be for security reasons, but unfortunately Hal is a truthful computer, and keeping a secret drives him loony-moons.
Poor Hal…but on the other hand this creates the only real actual plot kind of thing in the book, as Hal murders Poole and tries to kill Bowman. He also kills the folks in suspended animation before Bowman takes him down. Then it's back to descriptions of starscapes and not much happening.
Bye Hal. We miss you.
Bowman gets to Japaetus, and finds an even bigger slab on the moon. He figure he might as well check it out since the mission has gotten all messed up and he's going to die anyway before another ship can come out and rescue him. So he space-pods down to the giant slab, which is actually a Star Gate. It sucks him in and spits him out on the other end of the galaxy. (Insert sucking and spitting noises here. Cover your mouth, Star Gate.)
Bowman sees lots of cool astronomical stuff before his pod lands in a hotel room the aliens set up to make him feel comfortable. Then the aliens turn Bowman into a super space baby, and he zips back to earth. Bet that was not what you thought was coming.
Back on earth, he is either attacked by nuclear weapons or else gets back just in time to stop a nuclear war, it's not exactly clear which. Either way, he zaps the nuclear weapons with his space baby powers and then floats out there in space baby contemplation, trying to figure out what to do with the earth.
So basically the whole novel is a lengthy origin story to explain the coming of super space baby.