In literature, the climax is the point at which the narrative tension reaches its...
The word "climax" is defined by the Oxford English Dictionary as:
The culmination, peak, or apex of something; the most important or exciting part of a film, contest, etc., usually happening near the end.
In literature, the climax is the point at which the narrative tension reaches its highest pitch and can no longer be sustained. The tension is often released by some kind of crisis, defined as "a vitally important or decisive stage in the progress of anything; a turning-point," which forces the action to a resolution.
The Iliad begins with "the rage of Peleus' son Achilles," and climaxes when that rage results in the death of Hector, prince of Troy.
The poem starts at the end of a very long war, in which neither side has made any significant progress for some time. All the participants are tired and disaffected. A minor disagreement between Achilles and Agamemnon turns into a major falling-out with huge ramifications, as Achilles, foremost among warriors, withdraws from the battlefield. The Achaeans lose their best man and their morale is badly damaged.
Meanwhile the Trojans, led by Hector, take notice of Achilles' absence. They have been under siege for a decade, and unable to throw off the Achaeans largely due to Achilles' prowess in battle. With Achilles no longer in play, Hector sees an advantage for the Trojans and presses it for all it's worth. The war, which has been at a stalemate, comes roaring back to life, and the Achaeans are beaten back from the walls of Troy as far as their own ships. When the Trojans set the Achaean ships on fire, trapping the soldiers on the shoreline, Achilles' dearest friend, Patroclus, joins the battle, dressed in Achilles' armor in an attempt to rally the troops and fend off the Trojan advance. Hector kills Patroclus in combat.
Achilles is devastated. His anger towards Agamemnon forgotten, he reenters the battlefield and cuts a bloody swathe across it, furious for revenge. The Achaeans take courage from Achilles' presence and throw the Trojans back, killing hundreds. The Trojan army retreats behind the city walls—all except Hector, who stands before Troy's gates and waits for Achilles to come for him. His father, King Priam, begs him to come within the walls, saying Achilles has already killed so many of Hector's brothers; he, Priam, cannot stand to lose another son. Hector refuses to yield, however, and holds his ground until Achilles is almost upon him, determined to kill Achilles or be killed by him rather than retreat:
...Better by far for me
to stand up to Achilles, kill him, come home alive
or die at his hands in glory out before the walls.
...So [Hector] wavered,
waiting there but Achilles was closing on him now
like the god of war...
But as Achilles draws closer, Hector's nerve fails, and he takes off running:
Hector looked up, saw him, started to tremble,
nerve gone, he could hold his ground no longer,
he left the gates behind and away he fled in fear and
Achilles went for him, fast, sure of his speed
as the wild mountain hawk ...
breakneck on in fury
with Hector fleeing along the walls of Troy,
fast as his legs would go.
... they raced, one escaping, one in pursuit
and the one who fled was great but the one pursuing
greater, even greater-their pace mounting in speed
since both men strove, not for a sacrificial beast
or oxhide trophy, prizes runners fight for, no,
they raced for the life of Hector breaker of horses.
All the action of the war these past ten years is now honed to the knifepoint of these two men, Achilles and Hector, confronting each other at last. They are the proxies of their respective armies, Achaean and Trojan, and the outcome of their confrontation will determine the final course of the entire war. These men are individually more important than all the other characters in the poem; the gods' interference in the war is centred largely around the two of them. This climactic scene of Achilles running towards Hector before the city gates is a depiction of an unstoppable force meeting an (almost) immovable object—the results, more than climactic, are cataclysmic. Achilles catches up to Hector and kills him savagely, lashing Hector's corpse to his chariot and dragging it around the walls of Troy to the horror of the citizens watching from above. The spirit of the Trojan army is destroyed.
The final few books of the poem comprise the denouement, describing how Achilles' rage gives way to grief for Patroclus, which allows him to empathise with the grief King Priam feels for his dead son, Hector. Achilles returns the body of Hector to the Trojans, and the poem ends.
(All quotes are from the Robert Fagles translation of the Iliad.)