Ajax is the son of Telamon, the old king of Salamis at the time of the Trojan War. In literature, he is referred to as the Telamonian Ajax, Ajax of Salamis, or simply as Big Ajax.
During the Trojan War, Ajax was the bulwark of the Greek army. He was the largest of the Greeks, and often he fought next to his brothers, the Oilean Ajax, or "Little" Ajax, and Teucer. In single combat, he battled Hector to a draw. Then, the two exchanged weapons. Hector gave him his sword, and Ajax his girdle. When Agamemnon sent an embassy to the sulking Achilles, Ajax was a member. During this time when the other great kings were hurt, Ajax continued to fight every day, holding off the Trojans as best he could. During every advance retreat, Ajax, with his huge shield, covered the Greek soldiers.
When Achilles died from an arrow shot by Paris, there was a huge battle for the corpse. Eventually, Odysseus and Ajax managed to carry it from the battle and back to the ships. Both Odysseus and Ajax claimed his armor.
In order to prevent bloodshed, Odysseus proposed to let the other kings decide. Ajax agreed. He was certain that he would win, because he had fought so hard for so long, and he was closer to Achilles than Odysseus had been.
The two pleaded their cases before the judges, but Odysseus was by far the better speaker. Ajax could not match his eloquence. The armor then was awarded to Odysseus, and Ajax fled the camp in anger.
That night, outside the Greek camp, Ajax planned his revenge. He snuck back into the camp, and began to kill every Greek he saw. Athena had sent a madness upon him, however, and he was actually cutting down a herd of livestock. In the morning, he gloated over his victories, but eventually began to realize what he has done. Completely disgraced, Ajax fell upon his own sword, which had been given to him by Hector when he was at the height of his glory.
After his death, Agamemnon and Menelaus refused to allow Ajax to be buried. After Odysseus spoke on his behalf, however, the two relented. Later, when Odysseus visited the underworld in Book 11 of the Odyssey, Ajax, still angry in death, was the only Greek to refuse to speak to him.