Why does a conflict take place between Agamemnon and Achilles in the Iliad ?

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.