Note: I've tried to use accurate Greek transliteration throughout, hence Achilleus, Hektor, and Olympos, rather than Achilles, Hector, and Olympus. The exception is Hera rather than Here.
Peleus wooed the nymph Thetis, and they were to be married. All the gods were invited to the wedding, save Eris (they must have figured she'd wreck the party, being the goddess of discord). She still managed to make trouble, though, tossing a golden apple into the hall. The apple was emblazoned with the words "for the fairest." Naturally, Hera, Athena, and Aphrodite each assumed it was for herself--the three must have conked heads bending down to pick it up.
To quell he commotion that ensued, Zeus decreed that the fairest among mortal men should judge the beauty contest--whomever he chose would receive the apple. Paris, a prince of Troy and shepherd by trade, was chosen as the judge. Hermes came to him and told him the tale, and laid the duty of choosing the fairest goddess upon this stunned, if good-looking, young man.
Meanwhile, the princes of Greece were busy wooing the fairest of mortal women, Helen. She was so beautiful that it filled them all with a madness to have her. Somebody realized that, no matter whom she chose, her beauty could cause everlasting strife if some action were not taken. It was decided that the all the suitors--including the son of Peleus and Thetis, Achilleus--would swear a solemn oath to support Helen's decision and come to the aid of her chosen husband if needed. If ever anyone challenged her rightful man, they swore, the Greek nobility would unite and make war upon him...should Helen be abducted, they would do whatever it took to win her back for her husband. Helen chose Menelaus, king of Sparta. Not the strongest Greek leader, but he must have been charming.
Back on Olympos, the three contestant goddesses meditated on ways and means to get the apple. In their pride and power, none of them were above bribing their way to victory. Each offered Paris a boon in exchange for his vote. Hera offered him power, Athena wisdom, and Aphrodite the most beautiful woman in the world. Maybe it was a life spent among sheep that led him to choose the woman, despite the fact that power or wisdom would have been more practical--Paris chose Aphrodite. She had neglected to mention that the most beautiful woman in the world was already married.
Aphrodite arranged for Paris to be the guest of Menelaus, who was sharing a happy home with the amazingly attractive Helen. Using his own natural charms and aided by Aphrodite's wiles, Paris seduced Helen and took her back to Troy. Thus the oath was invoked and the princes of Greece prepared for war.
Some time had passed, however, and not everyone was eager to go off and get killed over this woman they used to lust after. Most of them already had one or two wives, and in any case Helen was beyond their reach. Their only reason to fight was an oath sworn hastily in the heat of courtship, and many now regretted it. Menelaus' brother Agammemnon was the stronger leader, and he went about the land rallying the princes, often against their wills.
Crafty Odysseus, for example, pretended to be mad when Agammemnon came knocking. He stripped himself naked and took to the fields, pulling a plow and braying like an ox. Agammemnon saw through this charade, and sat with his counselors nearby, discussing tactics. Odysseus' pride overcame him, and he couldn't stop himself from speaking out, correcting Agammemnon's faulty strategy, telling him how he ought to arrange his battle. Thus Odysseus was caught.
Greece's strongest warrior, young Achilleus, was also loath to go. When Agammemnon and train came to his palace, he disguised himself as a woman. Agammemnon was ready to give up and leave when Odysseus figured out the ploy. He ordered all the women in the palace to gather in the great hall, and caused two chests to be brought forth. One was filled with fine jewelry, and one with weapons. They noted that all the true women gathered about the first chest, but one "woman" could not resist the other. Betraying his disguise, Achilleus couldn't resist fondling the daggers and swords and axes. So he too was caught.
When all the host had been gathered at Agammemnon's citadel in Argos, they prepared to sail across the sea to Asia Minor where Troy lay. The gods, however, would not give them a favorable wind. Someone looked at some bird entrails and determined that a grave sacrifice was in order. Despite the severe protestations of his wife, Agammemnon opted to sacrifice his daughter to succor the gods' favor. The bloodthirsty gods, though they would punish Agammemnon in the long run, granted his fleet the wind it needed to sail to Troy.
Thus began an extremely violent conflict of dubious merit. For ten years, the Achaeans (as the Greeks were known, after the Achaea region of Greece) troubled Troy. King Priam of Troy, Paris' father, had another son more honorable who took up the defense of the town. He was known as Hektor, and gave the Achaeans much grief. "Manslaughtering Hektor" he was called, and according to Homer, the Greeks in their hundreds before him dropped and died. But he was fair, and honorable, and had an adoring wife named Andromache and a beautiful baby boy who was frightened when his father wore his bloodstained helm.
Priam's other son, Paris abductor of Helen, also participated in the fight from time to time when he wasn't attending to his woman's sexual needs. Usually he got himself into trouble and had to be rescued by Aphrodite.
The war raged on and on, and many noble princes met their dooms. The gods themselves participated, sometimes getting wounded even, such was the caliber of the heroes of those days. Though Olympos was divided in the conflict (many gods had children on one side or the other), it was ordained by Zeus that Troy would fall. But not before much horror and bloodshed.
Achilleus' best friend (some say lover, but this all happened before the swinging classical era) Patraklos was slain by Hektor. As it happened, Achilleus and Agammemnon had had a fight over a woman. Agammemnon had been forced to give up a priest's daughter he had taken when the priest's offer of ransom was refused and he called upon Apollo to smite the Greek army until Agammemnon relented. Apollo, the god of healing but also of disease, inflicted a plague upon the Greek livestock, and rather than watching all their cows and oxen die, Agammemnon grudgingly gave up the girl. Subsequently, and unfairly, he took Achilleus' war-bride as compensation. Achilleus bitterly withdrew from the fight, sending Patraklos out in his stead. Patraklos distinguished himself, hacking many a Trojan, until he met Hektor and was bested. Only then did Achilleus' hatred of Hektor overcome his anger with Agammemnon, and he rejoined the war.
After much smiting, Achilleus finally caught Hektor and struck him down. Not content with mere victory, he bored holes in Hektor's heels and strung a cord through them, so he could drag the body behind his chariot around the walls of Troy for several days. Eventually, Priam came under flag of truce to buy the body of Hektor from Achilleus and dispose of it honorably. Not long after, a cowardly--some say poisoned--arrow from the bow of Paris found Achilleus, and the greatest fighter among the Achaeans died in his turn.
In the tenth year of gruesome fighting, Odysseus had a plan to end the horrible stalemate. Making as if to leave, the Greeks constructed a huge wooden sculpture of a horse. Their ships pulled out, leaving the horse sculpture as an ostensible offering to Hera (who, by the way, was on the Trojan side throughout the war). Finding the Greeks gone, the Trojans decided to bring the horse into their town. This was after some debate; a few Trojans felt that this was one gift horse that deserved to be burned or rolled into the sea. The Trojan prophetess Kassandra--who was cursed with the gift of true divination that no one would ever believe--said it would be the end of Troy to bring the Greek gift home.
Odysseus and other picked warriors from the Greek host were, of course, hidden within the belly of the horse. They must have been nervous hearing talk of burning and drowning, but they had the last laugh as the Trojans wheeled the mighty structure inside their walls. The Greeks emerged that night, finding most of the town drunk with victory revelry, and went this way and that, slaying as they chose. They opened the gates, and the main Greek host, which had lain hidden in the vessels behind a nearby island, sauntered in. The sack was complete, and Priam was killed in his court by Achilleus' son.
Few Trojans escaped. Helen was returned to Menelaus without penalty, other than the fact that she was beholden to recount the tale to visitors, blaming herself as a slut. Hektor's son was, on Odysseus' advice, cast from the walls of Troy so that he might not grow to manhood and seek vengeance. Lust-driven Agammemnon took Kassandra home to be his consort; the pair were axe-murdered in the bath by Agammemnon's wife, who was not only feeling slighted by Agammemnon's doting on his new lover, but was also still more than peeved about the sacrifice of her daughter. Odysseus was punished for pride by Poseidon; it took him another ten years to get home, during which time all of his followers lost their lives while his wife and son had to contend with a host of pushy suitors looking to move in on his kingdom. Few happy tales resulted from this war.
At the time when this tale was sung, the Trojan War was the greatest armed conflict known to the bards. It's a story of immense power and tragedy, and one from which Western civilization could have learned a few things. Alas, we don't seem to have learned much.