Agamemnon did not stay in the good graces of either of the children of Leto for long. He soon incurred the wrath of her son Apollo, the brother of Artemis. In revenge, Apollo caused an outbreak of plague to lay the troops low.
Agamemnon and Achilles had received Chryseis and Briseis as prizes of war or war brides. Chryseis was the daughter of Chryses, who was a priest of Apollo. Chryses wanted his daughter back and even offered a ransom, but Agamemnon refused. Calchas the seer advised Agamemnon on the connection between his behavior toward the priest of Apollo and the plague that was decimating his army. Agamemnon had to return Chryseis to the priest of Apollo if he wanted the plague to end.
After much Greek suffering, Agamemnon agreed to the recommendation of Calchas the seer, but only on condition that he take possession of the war prize of Achilles -- Briseis -- as a replacement.
A minor point to think about: When Agamemnon had sacrificed his daughter Iphigenia, he hadn't required his fellow Greek aristocrats to give him a new daughter.
No one could stop Agamemnon. Achilles was enraged. The honor of the leader of the Greeks, Agamemnon, had been assuaged, but what about the honor of the greatest of the Greek heroes -- Achilles? Following the dictates of his own conscience, Achilles could no longer cooperate, so he withdrew his troops (the Myrmidons) and sat on the sidelines.
With the help of fickle gods, the Trojans began to inflict heavy personal damages on the Greeks, as Achilles and the Myrmidons sat on the sidelines. Patroclus, Achilles' friend (or lover), persuaded Achilles that his Myrmidons would make the difference in the battle, so Achilles let Patroclus take his men as well as Achilles' personal armor so that Patroclus would appear to be Achilles in the battlefield.
It worked, but since Patroclus was not so great a warrior as Achilles, Prince Hector, the noble son of Trojan King Priam, struck Patroclus down. What even Patroclus' words had failed to do, Hector accomplished. The death of Patroclus spurred Achilles into action and armed with a new shield forged by Hephaestus, the blacksmith of the gods (as a favor for Achilles' sea goddess mother Thetis) Achilles went into battle.
Achilles soon avenged himself. After killing Hector, he tied the body to the back of his war chariot, The grief-maddened Achilles then dragged Hector's corpse through the sand and dirt for days. In time, Achilles calmed down and returned the corpse of Hector to his grieving father.
In a later battle, Achilles was killed by an arrow to the one part of his body Thetis had held when she had dipped the baby Achilles into the River Styx to confer immortality. With Achilles' death, the Greeks lost their greatest fighter, but they still had their best weapon.