There is a very good summary here at enotes (the third link below.) The line of the story goes like this: Helen, the most beautiful woman in the world, is the wife of King Menelaus of Sparta. Menelaus' brother is Agamemnon of Mycenae, the mightiest king among the Achaians (the Greeks). Paris, a prince of the city of Troy (across the Aegean sea from Greece) steals Helen away from Menelaus and brings her to Troy. This provides the excuse for Agamemnon to wage war on Troy, with the "thousand ships" of all the kingdoms of Greece. At the battles before the city of Troy, the hero Achilles is the Greek's best hope; but he sulks, and will not fight, when a woman he won in battle is taken from him by the arrogant leader, Agamemnon. Patroclus, Achilles' best friend, falls in battle, which angers Achilles so that he returns to battle and finally kills the hero of Troy, Prince Hektor. He abuses Hektor's body, which grieves the Trojans. Priam, the king of Troy, goes on a secret midnight mission to the Greek camp, and begs Achilles to give back the body of his son. Achilles relents, and funeral games are held for Hector.
There is more to the story after the text of the Iliad (the action of the poem takes place over a period of only about fifty days during the tenth year of the war.) These are the main events in the poem itself, although the story of Helen's abduction takes place in the past.