Aboard a British ship called the Nellie, three men listen to a dude named Marlow recount his journey into Africa as an agent for "the Company," a Belgian ivory trading firm.
If you think "The Company" sounds super-sketchy, you're right: from the get-go, Marlow feels a nameless sense of dread about working for "The Company." (It doesn't help that the last guy to have held Marlow's position...was murdered.)
When Marlowe signs on to take this voyage, he sees a couple of old women knitting in the corner. They give him the heebie-jeebies. Then, when he gets to Africa, he meets a dude wearing starched, formal clothing despite the heat. He's deeply weirded out by this fancy-pants guy and by the camp in general—and things haven't even started to get nightmarish.
Marlow realizes that the Africans are kept as slaves, and many are dying from the brutality of the conditions. These Africans, he realizes, and "not inhuman." (Don't get excited; Marlow's hardly progressive here.)
As the bureaucracy of The Company moves at a molasses-like pace, Marlowe becomes entangled in a power struggle within The Company—middle management is trying to climb the ranks, and being especially slimy about it. He also starts hearing tell of a mysterious figure named Kurtz, a mad agent who's rumored to have become both a prisoner and revered as a god by the indigenous population living further down the Congo.
In fact, the more he hears about Kurtz, the more obsessed Marlow becomes. Who is this Kurtz? Why is he such a powerful figure? Why does everyone seem to either idolize him or loathe him?
Finally, after delays due to a broken-down (or possibly vandalized) steamship, Marlow is on his way to meet the enigmatic Kurtz. Aboard the steamship are cannibals who, thankfully, snack on some rancid hippo meat. The ship is forced to stop often: once to pick up wood (the pile of wood is accompanied with a note that says, essentially, "Proceed with caution"), once because of a mysterious fog bank, and once because of an attack—arrows strike the ship from the riverbank, and the helmsman is impaled with a spear.
When the riverboat arrives at Kurtz's camp, Marlow sees that the decoration of choice is posts topped with the severed heads of locals. Oh, that's not creepy at all.
Marlow's met by a weird Russian dressed like a clown. This "harlequin" informs Marlow that Kurtz is a god. He has apparently expanded the harlequin's worldview with his power and eloquence. Marlow, naturally, thinks that both this harlequin and Kurtz must be bonkers.
We finally meet Kurtz, who's so ill he's carried out on a stretcher. He looks, almost literally, like death. The natives in the camp want to attack the steamship, but Kurtz tells them to fall back. Although Kurtz tries to run away—or, rather, crawl away, because he's too weak to run—he finally agrees to being brought back to the mouth of the Congo.
Kurtz's health deteriorates: he goes blind, starts raving in a series of fever dreams, and gives Marlow a stack of papers and the instructions not to hand them over to "The Company." When he finally dies, his (famous) last words are "The horror! The horror!" Shortly after this, Marlow becomes super sick, but he pulls through.
When he returns to Europe, Marlow is disillusioned with both "The Company" and Europe in general. He refuses to hand over Kurtz's papers, possibly jeopardizing his career. He does, however, visit Kurtz's Belgian fiancée, in order to give her a few of Kurtz's letters. The fiancée is still mourning Kurtz, and asks anxiously about Kurtz's last words. Marlow lies, telling her that Kurtz said her name.
Yeah, that's a lot sweeter than muttering, "The horror! The horror!"