The conflict between Achilles and Agamemnon over the sex slave Briseis may seem somewhat trivial on the face of it. But to both these men with big egos, this is a matter of honor.
As far as Achilles is concerned, Briseis is his; she belongs to him like any war booty. Disturbing as it is to us today, it was quite common for warriors in ancient civilizations to take women as spoils of war and use them as slaves or concubines. Achilles is one of many such warriors who believes himself entitled to such a glittering “prize.”
So when Agamemnon appropriates Briseis for himself, Achilles is absolutely furious. He feels he's been disrespected and dishonored as a warrior, even though as king, Agamemnon is perfectly entitled to take whatever he wants from one of his subordinates. Even though Agamemnon has a perfectly valid reason for snatching Briseis from Achilles—he has to in order to replace his own slave whom he had to give up to appease the wrath of the gods—Achilles is so full of pride that he cannot accept what he regards as a personal affront to his dignity as a warrior.
Of course, Briseis's dignity doesn't enter into the equation at all. As both a woman and a slave, she doesn't really count in the hyper-masculine society depicted in the Iliad.
Further Reading
The quarrel between Agamemnon and Achilles which forms the principal subject of the first half of the Iliad was caused by Agamemnon taking Achilles's concubine Briseis. Agamemnon was forced to give up his own concubine Chryseis at the behest of Apollo, and his assumption that he could simply take Briseis instead so outraged Achilles that he withdrew to the ships and stopped fighting for Agamemnon.
Achilles declares that he loved Briseis, but it appears that he is more concerned with the insult to his honor than the actual loss of the girl. The precise status of the client kings who fought for the High King of Mycenae is a matter of dispute. They were semi-independent but clearly also owed some allegiance to Agamemnon. Agamemnon himself says that he is "more kingly" (the Greek word is "βασιλευτερος") than Achilles and that therefore Achilles should submit to him. This seems to have angered Achilles so much that even after Agamemnon sends an embassy, in book 9, with fabulous gifts (including many more concubines), Achilles still refuses to be reconciled.
This conflict emerges at the beginning of the poem and is crucial to the advancement the plot. It also marks an interesting parallel to the cause of the Trojan War itself: a dispute over Helen between Menelaus and Paris. After the Greeks sack a city allied with Troy, Agamemnon and Achilles each make off with a young woman. When Agamemnon is forced to give up his "prize," a girl named Chryseis, he determines to take Briseis, who has been awarded to Achilles. Achilles angrily protests and orders his men not to fight with the Greeks. He seeks the support of...
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Zeus, who aids the Trojans. They meet with great success, thus showing the importance of Achilles to the Greek cause. The disagreement, then, is really about honor. Agamemnon profoundly insults Achilles by taking Briseis from him, and since Achilles never really acknowledged Agamemnon's authority, he revolts. Even when Agamemnon agrees to return Briseis along with some gifts, Achilles still sulks in his tent. Only the death of Patroclus (disguised as Achilles himself) at the hands of Hector motivates Achilles to rejoin the fight. He avenges his friend's death by killing Hector, whose funeral marks the end of the poem.
Essentially, the conflict between the two is one rooted in pride and ego. Agamemnon is the leader of the Achaean empire and the one for whom expansion and conquest is of the most vital of importance. Achilles is the brave warrior and great fighter. The challenges between them is one where the former seeks to establish glory and power, but can only do so through the heroic valor of the latter. The collision between them ends up becoming which one should wield more power. Achilles is a warrior, one who believes in his own glory on the battlefield and his own sense of arete. He believes Agamemnon as one who can only reap the benefits of what other soldiers have done. For his part, Agamemnon believes that history remembers leaders and kings, and soldiers, like Achilles, have to understand that natural hierarchy. This becomes the root of their conflict, in that both believe in their own superiority and their own subjective notion of the pursuit of excellence. Both refuse to acquiesce to the other's vision and thus conflict arises between them.
I assume that you are asking about the conflict between Agamemnon and Achilles that takes place in Book I of this epic. In that case, the conflict comes about because Agamemnon wants to take Briseis away from Achilles. Briseis is a woman who has been given to Achilles as a war prize.
Agamemnon had also had a woman, Chryseis, given to him as a war prize. But her father was able to get the Achaeans to agree that she should be given back to him. When that happened, Agamemnon demanded that he should be compensated for the loss of his prize. The only compensation that he would agree to was Briseis.
When Briseis was taken from him, Achilles' honor was insulted and the conflict arose betwen him and Agamemnon.
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.
What is the conflict that starts the problems between Agamemnon and Achilles in the first place?
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.
Further Reading
What is the conflict that starts the problems between Agamemnon and Achilles in the first place?
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.