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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.
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