Obviously it would be cheating to cut and paste from Wikipedia, but why not get us to answer the question and just cut and paste what we write. Here's my contribution:
In Greek mythology, Hectōr (Ἕκτωρ), or Hektōr, was a Trojan prince and the greatest fighter for Troy in the Trojan War. As the first-born son of King Priam and Queen Hecuba, a descendant of Dardanus, who lived under Mount Ida, and of Tros, the founder of Troy, he was a prince of the royal house and the heir apparent to his father's throne. He was married to Andromache, with whom he had an infant son, Scamandrius (whom the people of Troy called Astyanax). He acts as leader of the Trojans and their allies in the defense of Troy, killing 31 Greek fighters in all. In the European Middle Ages, Hector figures as one of the Nine Worthies noted by Jacques de Longuyon, known not only for his courage but also for his noble and courtly nature. Indeed Homer places Hector as peace-loving, thoughtful as well as bold, a good son, husband and father, and without darker motives. When the Trojans are disputing whether the omens are favorable, he retorts: "One omen is best: defending the fatherland."
Etymology
In Greek, Héktōr is a derivative of the verb ékhein, archaic form hékhein, "to have" or "to hold". Héktōr, or Éktōr as found in Aeolic poetry, is also an epithet of Zeus in his capacity as "he who holds [everything together]". Hector's name could thus be taken to mean "holding fast".
Greek mythology
Greatest warrior of Troy
Hector Admonishes Paris for His Softness and Exhorts Him to Go to War by J.H.W. Tischbein (1751–1828)
According to the Iliad, Hector did not approve of war between the Greeks and the Trojans.
For ten years the Achaeans besieged Troy and their allies in the east. Hector commanded the Trojan army, with a number of subordinates including Polydamas, and his brothers Deiphobus, Helenus and Paris. However, by all accounts Hector was the best warrior the Trojans and all their allies could field, and his fighting powers were admired by Greeks and his own people alike.
Diomedes and Odysseus, when faced with his attack, described him as what Robert Fagles translated as an 'invincible headlong terror', and a 'maniac'.
Duels with warriors
Duel with Protesilaus
In the Iliad, Hector's exploits in the war prior to the events of the book are recapitulated. He had fought the Greek champion Protesilaus in single combat at the start of the war and killed him. A prophecy had stated that the first Greek to land on Trojan soil would die. Protesilaus, Ajax and Odysseus thus would not land. Finally, Odysseus threw his shield out and landed on that, and Protesilaus jumped next from his own ship. In the ensuing fight, Hector killed him, fulfilling the prophecy.
Ajax and Hector exchange gifts (woodcut in Andreas Alciatus, Emblematum libellus, 1591)
Duel with Ajax
At the advice of his brother Helenus (who also is divinely inspired) and being told by him that he is not destined to die yet, Hector manages to get both armies seated and challenges any one of the Greek warriors to single combat. The Argives are initially reluctant to accept the challenge. However, after Nestor's chiding, nine Greek heroes step up to the challenge and draw by lot to see who is to face Hector. Ajax wins, and fights Hector to a stalemate for the entire day. With neither able to achieve victory, they express admiration for each other's courage, skill, and strength. Hector gives Ajax his sword, which Ajax will later use to kill himself. Ajax gives Hector his girdle, which will later be used to attach Hector's corpse to Achilles' chariot by which he is dragged around the walls of Troy.
The Greek and the Trojans make a truce to bury the dead. In the early dawn the next day the Greeks take advantage of it to build a wall and ditch around the ships. Zeus is watching in a distance.
Duel with Achilles
Another mention of Hector's exploits in the early years of war was given in the Iliad book 9. During the embassy to Achilles, Odysseus, Phoenix and Ajax all try to persuade Achilles to rejoin the fight. In his response, Achilles points out that while Hector was terrorizing the Greek forces now, and that while he himself had fought in their front lines, Hector had 'no wish' to take his force far beyond the walls and out from the Skiaian Gate and nearby oak tree. He then claims, 'There he stood up to me alone one day, and he barely escaped my onslaught.'
A 2004 film version of Troy has Achilles slaying Hector following a duel, whereas in the Iliad it is rather different. Hector remains outside the walls, while his army flees into the city. As Achilles approaches, Hector stands his ground, fights and ...