Ajax is known for his size and strength, so much so that the tag line of a popular cleaning product was "Ajax: Stronger than dirt." There were actually two Greek heroes in the Trojan War named Ajax. The other, physically much smaller Ajax is the Oilean Ajax or Ajax the Lesser.
Ajax the Greater is depicted holding a large shield that is compared with a wall.
Ajax the Greater was the son of the king of the island of Salamis and the half-brother of Teucer, an archer on the Greek side in the Trojan War. Teucer's mother was Hesione, sister of the Trojan King Priam. The mother of Ajax was Periboea, daughter of Alcathus, son of Pelops, according to Apollodorus III.12.7. Teucer and Ajax had the same father, Argonaut and Calydonian boar hunter Telamon.
The name Ajax (Gk. Aias) is said to be based on the appearance of an eagle (Gk. aietos) sent by Zeus in response to Telamon's prayer for a son.
Ajax and the Achaeans
Ajax the Greater was one of the suitors of Helen, for which reason he was obliged by the Oath of Tyndareus to join the Greek forces in the Trojan War. Ajax contributed 12 ships from Salamis to the Achaean war effort.
Ajax and Hector
Ajax and Hector fought in single combat. Their fight was ended by the heralds. The two heroes then exchanged gifts, with Hector receiving a belt from Ajax and giving him a sword. It was with the belt of Ajax that Achilles dragged Hector.
When Achilles was killed, his armor was to be awarded to the next greatest Greek hero. Ajax thought it should go to him. Ajax went mad and tried to kill his comrades when the armor was awarded to Odysseus, instead. Athena intervened by making Ajax think cattle were his former allies. When he realized he had slaughtered the herd, he committed suicide as his only honorable end. Ajax used the sword Hector had given him to kill himself.
The story of the madness and disgraced burial of Ajax appears in the Little Iliad. See: "Ajax's Burial in Early Greek Epic," by Philip Holt; The American Journal of Philology, Vol. 113, No. 3 (Autumn, 1992), pp. 319-331.
Even in his afterlife in the Underworld Ajax was still angry and wouldn't speak with Odysseus.