The Odyssey by Homer is the story of the return of Odysseus from the Trojan War. Thus the significance of the Trojan War is that it, especially as recounted in The Iliad by Homer, acts as a sort of prequel to The Odyssey. Many of the events that happen in the process of the return of Odysseus are dependent on prior events in the Trojan war and the alliances and hostilities among people and gods ensuing from that war. Both Athena's patronage of Odysseus and Poseidon's enmity to him spring in part from the events of the war.

What is the significance of the Trojan War outside of Homer's Odyssey?

The Trojan War is in its tenth grueling year when The Iliad takes place, which is significant because it's a catalyst for the desperation and in-fighting amongst the soldiers. The Trojans are still fending off the Achaians, and the Achaians have yet to get Helen back (their "official" goal), or ravage Troy for its riches (their other, more lucrative goal). In short, everybody's feeling stuck and frustrated. The Trojans can't quit because Troy will fall. The Achaians can't quit because they've invested ten years of bloodshed. If they give up now, what was that all for?

The war also drags on because Achilleus enlists the help of his mom, Thetis, a god, and all of her god-friends. The Achaians have the Trojans on the ropes, for example, and then Thetis calls in a favor to Zeus. Ouch.

This war without end influences the characters' actions. They get frustrated; they take risks, and this sets off the chain of events that drives the poem's narrative. Achilleus gives up his girlfriend in a deal with Agamemnon and then gets so mad about it that he refuses to fight. Patroklos then goes into battle in Achilleus' place—even going so far as to wear Achilleus' armor—and gets killed by Hektor (with help from Apollo; there's those meddling gods again). When Achilleus hears what happened to Patrokolos, he kills Hektor.

Hektor's funeral wraps things up for The Iliad, but it's clear that Achilleus is going to die, too, and Troy will fall to the Achaians. By the end, the seemingly infinite nature of the Trojan War wears everyone down to dust.