No single episode in the Odyssey does more narrative work than the Cyclops encounter. It is a clever escape story, a wordplay joke, and — thanks to one very human lapse in judgement — the direct cause of everything that goes wrong for the rest of the epic.
Trapped in the cave of Polyphemus, a Cyclops and son of Poseidon, Odysseus and his surviving men watch several companions eaten before devising their escape. Odysseus gets Polyphemus drunk on unusually strong wine and, when asked his name, answers Outis — "Nobody." That night, the men drive a fire-hardened stake into Polyphemus's single eye, blinding him. When his screams bring the other Cyclopes to his cave entrance asking what's wrong, Polyphemus shouts back that "Nobody" is attacking him — and, taking this literally, the others simply leave, assuming he's suffering some divine affliction rather than an actual attack.
Blind but still guarding the cave's entrance, Polyphemus feels along the backs of his sheep as he lets them out to graze each morning, checking that no man rides atop them. Odysseus's solution is to strap his men underneath the bellies of the rams instead — invisible to Polyphemus's searching hands — allowing the entire party to slip out unnoticed as the flock passes through the cave mouth.
The mistake that costs ten years: safely aboard ship and pulling away, Odysseus cannot resist gloating — he shouts his true name back at the shore, unable to leave Polyphemus without knowing exactly who defeated him. This single act of pride gives the blinded Cyclops exactly what he needs: a name to pray with. Polyphemus calls on his father Poseidon for vengeance, and the resulting curse is the direct cause of the storms, delays and disasters that stretch Odysseus's journey home into a full decade.
The engine of the whole epic: this single moment — pride overriding good sense at the exact wrong instant — is what turns a straightforward sea voyage into the ten-year odyssey the epic is named for.