What is the conflict that starts the problems between Agamemnon and Achilles in the first place?
It all starts when Agamemnon takes Chryseis as his own personal slave and concubine. She is a very beautiful woman; indeed, Agamemnon proclaims her even more beautiful than his wife, Clytemnestra. Not surprisingly, Agamemnon is reluctant to part with her, unwilling even to hold her for ransom. This creates a problem for the Achaeans as Chryseis's father is a priest of Apollo. When Agamemnon pointedly refuses to hand Chryseis back to her father, Apollo sends down a terrible plague which decimates the Achaean ranks. There's nothing for it; Agamemnon has no choice but to return Chryseis to her father.
But this creates an additional problem for the Achaeans. Agamemnon had set his heart on having Chryseis all to himself. So now he needs to find a replacement; and he finds her in the form of Briseis, she of the lovely cheeks. There's just one problem: she belongs to Achilles, the most fearsome warrior among the entire Achaean forces. Not surprisingly, Achilles is furious by what he perceives as a slight to his honor. Although Agamemnon may be a king, Achilles is a demi-god, son of Thetis, the sea-nymph. As such, he regards himself as a class apart from mere mortals such as Agamemnon.
Achilles, in an almighty sulk, retreats to the confines of his tent. His honor has been impugned, and he'll refuse to return to battle, even when his comrades are being slaughtered. It'll take yet another attack which he regards as personal--the killing of his close friend Patroclus--to get him to come out of his self-imposed exile and throw himself once more into the sting of battle.
This is a good question and one that is essential to explore, if you are going to understand the work. Here is the context:
After some victory among the Greeks, both Agamemnon and Achilles were awarded prizes (women as part of the spoils of war). Agamemnon obtained a woman named, Chryseis, and Achilles obtained a woman named, Briseis. However, Agamemnon had to give his "prize" back, because she was the daughter of the priest of Apollo. The god, Apollo, was attacking the Greeks with great fury and the only way to stop this was to give back the daughter of the priest of Apollo.
When this happend, Agamemnon wanted to take Briseis away from Achilles. This act was due to his pride, but more importantly it slighted the pride and honor of Achilles, which was paramout in Greek culture. For this reason, Achilles was filled with rage and decided not to fight for the Greeks. This caused the balance of power to favor the Trojans.
In terms of the rage of Achilles, here is what Homer says:
"Rage—Goddess, sing the rage of Peleus’ son Achilles,
murderous, doomed, that cost the Achaeans countless losses,
hurling down to the House of Death so many sturdy souls,
great fighters’ souls, but made their bodies carrion,
feasts for the dogs and birds,
and the will of Zeus was moving toward its end.
Begin, Muse, when the two first broke and clashed,
Agamemnon lord of men and brilliant Achilles."
In many ways, the Iliadis really about the rage of Achilles.
The essential conflict in the Iliad is one of honor, a fundamental issue in the warrior culture portrayed in the poem. The poem begins by giving the back story to this conflict. Agamemnon, after sacking the Greek city of Thebes, is forced by the anger of Apollo to return a young woman he claimed as a prize of war. This girl was the daughter of a priest to Apollo, and Agamemnon's initial refusal to return her, even for ransom, brought destruction down on the Achaeans.
When Agamemnon reluctantly returns her, he claims a girl taken by Achilles as compensation. Achilles has to give her up (the goddess Athena restrains him from resisting) but he is enraged, and refuses to fight against the Trojans. His troops are the best in the Greek army, and Achilles is its greatest warrior, and his refusal to participate is a terrible loss. But he also conspires with his mother, the nymph Thetis, to persuade Zeus to favor the Trojans in the conflict. So Agamemnon's arrogant decision to claim Briseis sparks a feud with Achilles that lasts until his friend Patroclus is killed by Hector. The "anger of Achilles son of Peleus" brings great misfortune to the Greek army.