The woman given to Achilles is named Briseis. She is given to Achilles as a prize (reward for his role in the fighting when the Achaians sack the city of Thebe) but later King Agamemmnon takes her away from Achilles and keeps her for himself.
When Agamemmnon does this, it causes one of the more famous episodes in the Iliad, as Achilles refuses to fight anymore and instead goes and sulks in his tent. He does this both because he is angry at having had his prize taken away and because he has fallen in love with her.
Eventually, Agamemmnon gives Briseis back to Achilles and Achilles consents to resume fighting for the Greeks.