The quarrel between Achilles and Agamemnon in Book 1 of Homer's Iliad sets in motion a chain of events that will affect the tenth and final year of the legendary Trojan War.
In the first nine years of the war, little had taken place because the Trojans had realized in the early days of the war that they were no match for Achilles on the open battlefield.
Once Achilles leaves the fighting, though, this gives the Trojans the courage to leave the safety of their city walls and come out to fight. Hector inflicts serious damage on the Greek fortifications on Troy's shores and even threatens to set fire to the Greek fleet. The threats to the Greek ships causes Achilles' best friend Patroclus to want to enter battle. In Iliad 16, Patroclus' death, at the hands of Hector, causes Achilles to return to the fighting and kill Hector (Iliad 22). Achilles then holds funeral games for Patroclus (Iliad 23) and ransoms back Hector's body to Priam (Iliad 24).
We should note, however, that the Trojan War does not end at this point. We should also note that Hector's death did not mean the defeat of the Trojans. The Trojans would get reinforcements. We should also keep in mind that Achilles himself did not live to see the fall of Troy. He is killed later in the tenth year by Paris/Alexander.
The quarrel sets the stage for the rest of the epic. It actually creates the main theme of the epic. Many people feel that this epic is about the Trojan War, but in actuality this is an epic poem about the wrath of Achilles. When Agamemnon takes away his war prize (a woman named Briseis) Achilles decides to sit out of the Trojan War. This sets forth a chain of events that ultimately leads to death of the great Trojan warrior Hektor.
Since Achilles, the greatest warrior in the Greek army, was sitting out, the Trojans were able to take the upper hand in the war. However, once Hektor killed Achilles' best friend, Patroclos, Achilles decides to seek revenge and re-enter the fray. Achilles' becomes blood drunk and almost single handedly drives the Trojans back within the walls of their city. Achilles kills Hektor on the battlefield, thus seeking revenge and ending his rage.
The quarell in Book I begins his rage, and it is this rage that ultimately leads to the defeat of the Trojans by the Greek army.