Ajax is the son of Telamon—who was the son of Aeacus and grandson of Zeus—and Telamon's first wife, Periboea. Through Telamon he is the elder half-brother of Teucer, the renowned archer, and through his uncle Peleus (Telamon's brother) he is the cousin of Achilles.

The etymology of his given name is ultimately uncertain. It derives from Archaic Greek Αἴϝᾱς (Aíwās) and is comparable to Etruscan aivas, with which it appears to share common roots; further origin is unclear, though Αἴϝᾱς has been derived from Proto-Hellenic *aiweí, itself from Proto-Indo-European *h₂ey- ("vital energy, life"). By folk etymology the name was said to come from the root of aiazō (αἰάζω), "to lament," rendering it "one who laments; mourner." Hesiod gave a different folk etymology in The Great Eoiae, where Heracles prays to Zeus that a son be born to Telamon and Eriboea: Zeus sends an eagle (aetos) as a sign, and Heracles bids the parents name their son Ajax after the eagle.

Many illustrious Athenians—including Cimon, Miltiades, Alcibiades and the historian Thucydides—traced their descent from Ajax. On an Etruscan tomb at Bologna (5th century BC) an inscription reads aivastelmunsl, meaning "[family] of Telamonian Ajax."

According to Dares the Phrygian, Ajax was "powerful. His voice was clear, his hair black and curly. He was perfectly single-minded and unrelenting in the onslaught of battle." Homer's Iliad describes him as of great stature and colossal frame, the strongest and tallest of all the Achaeans. Known as the "bulwark of the Achaeans," he was trained—like his father Telamon and like Peleus, the father of Achilles—by the centaur Chiron.

Fearless, strong and intelligent in combat, Ajax wielded an enormous shield made of seven cowhides faced with a layer of bronze. He sustained no wounds in any of the Iliad's battles and received little divine aid; in Book 13, Poseidon renews his strength with a touch of his staff. Unlike Diomedes, Agamemnon and Achilles, Ajax fought chiefly as a defensive warrior, protecting the Greek camp, the ships and the body of Patroclus, and frequently covering Achaean retreats. Despite being among the deadliest of warriors, he has no aristeia (a sustained sequence of offensive heroic exploits) in the poem.

Ajax's strength and courage are most prominent in two encounters with Hector. In Book 7 a lottery selects Ajax to duel Hector for a whole day; Ajax gains the advantage, wounding Hector with his spear and felling him with a great stone, but Hector fights on until heralds, acting under Zeus's direction, declare a draw. The two exchange gifts: Ajax gives Hector his "war-belt, glistening purple," and Hector gives Ajax his "silver-studded sword."

Their second confrontation comes when Hector breaches the camp. In Book 14 Ajax hurls a giant rock at Hector that nearly kills him; in Book 15, after Apollo restores Hector's strength, Ajax wields an enormous spear and leaps between the ships, almost single-handedly repelling the Trojan armies. In Book 16 they clash again; Hector disarms Ajax (though leaves him unharmed), and Ajax retreats once he sees that Zeus favors Hector. The Trojans manage to set one Greek ship ablaze, nearly ending the war. Ajax kills many Trojan lords, among them Phorcys.

Ajax often fought alongside his brother Teucer, the famed archer: Ajax sheltered Teucer behind his shield while Teucer picked off Trojans. With Achilles absent because of his quarrel with Agamemnon, Book 9 sends Ajax, Odysseus and Phoenix to Achilles' tent to seek reconciliation; though Ajax speaks earnestly and is warmly received, he fails to persuade Achilles to return.

After the death of Patroclus, Hector tries to take the body, but Ajax—aided by Menelaus—drives off the Trojans and recovers the corpse on his chariot, although the Trojans have already stripped Achilles' armor from it. Ajax's prayer to Zeus to lift the fog over the battlefield, so the Greeks might fight or die in the light of day, became proverbial. According to Hyginus, Ajax killed twenty-eight defenders of Troy in all.

As the Iliad ends, Ajax and most of the Greek warriors are still alive. When Achilles is killed by Paris with Apollo's aid, Ajax and Odysseus fight off the Trojans to recover and bury his body beside that of Patroclus.

Both heroes then claim Achilles' divine armor, forged by Hephaestus, in recognition of their deeds, and a contest is held to decide who deserves it. Ajax argues that his strength and accomplishments—above all his defense of the ships against Hector—earn him the magical protection, but Odysseus proves the more eloquent and, with Athena's help, the council awards him the armor. Overcome by his own grief, Ajax drives his sword through his chest and kills himself.

In the Little Iliad, Ajax falls into a rage-driven madness on losing to Odysseus and slaughters the Greek cattle; on regaining his senses he kills himself in shame. In Sophocles' tragedy Ajax, the insult drives him to plot the murder of Agamemnon and Menelaus, but Athena clouds his mind so that he falls instead upon a flock of sheep, taking them for the Achaean leaders. Coming to his senses, covered in blood, he chooses death over dishonor, using the very sword Hector had given him. From his blood springs a red flower—as at the death of Hyacinthus—bearing the letters "Ai" on its petals, an expression of lament. His ashes were laid in a golden urn at the Rhoetean promontory by the entrance to the Hellespont.

Ajax's half-brother Teucer stood trial before their father for failing to bring back Ajax's body or weapons; acquitted of responsibility but found guilty of negligence, he was disowned and forbidden to return to Salamis. Homer is vague about the manner of the death but attributes it to the loss of the armor dispute: when Odysseus visits Hades, he begs Ajax's soul to speak, but Ajax, still resentful of the old quarrel, refuses and descends silently into Erebus.

Like Achilles, post-Homeric legend has Ajax survive death on the island of Leuke at the mouth of the Danube. As grandson of Aeacus and great-grandson of Zeus, he became the tutelary hero of Salamis, with a temple and image there, honored in the festival Aianteia, at which a couch displayed the hero's panoply. After the Athenians acquired Salamis the family connection to Aeacus was emphasized—Solon was said to have inserted a line into the Iliad (2.557–558) to support Athens's claim to the island—and Ajax became an Attic hero, worshipped at Athens with a statue in the marketplace and giving his name to the tribe Aiantis. Pausanias reported that a gigantic skeleton, with a kneecap about five inches across, appeared near Sigeion on the Trojan coast and was identified as Ajax's remains.

In 2001 Yannis Lolos began excavating a Mycenaean palace near the village of Kanakia on the island of Salamis, which he proposed may have belonged to the mythological Aiacid dynasty. The multi-story structure covered about 750 m² (8,100 sq ft) with roughly thirty rooms and appears to have been abandoned at the height of Mycenaean civilization—around the probable period of the Trojan War.

By tradition, Ajax was held to have aided the Greeks at the Battle of Salamis. In his honor the festival Aianteia was held, at which young Athenian ephebes traveled each year to Salamis to compete with the local youth in foot-races and boat-races.