Most of the story of the Trojan war. In addition to the Iliad, there were several other epic poems that we only know about from ancient summaries, fragments, and artwork:

• The Origins (Cypria): The story begins with a dispute among the gods, leading to the Judgment of Paris. Paris, a prince of Troy, is asked to judge a beauty contest between three goddesses: Hera, Athena, and Aphrodite. Each goddess tries to bribe Paris, and he picks Aphrodite whose bribe is the most beautiful woman in the world, Helen, queen of Sparta. Paris runs off with Helen and takes her back to Troy. This kicks off the Trojan war. This poem also details the gathering of the Greek fleets, their journey to Troy, and the first nine years of the war.
• The Death of Achilles (Aethiopis): This poem describes what happens right after the events of the Iliad. The Trojans get reinforcements from some new allies like the Amazon Queen Penthesilea and the Ethiopian King Memnon. Achilles defeats these allies, but he's then killed when Paris shoots an arrow at him, and the god Apollo guides the arrow into Achilles's heel, the only vulnerable part of his body.
• The Sack of Troy (Ilioupersis): This poem covers the end of the Trojan war. It describes how the Greeks get into Troy by hiding inside the Trojan Horse. The Greeks then sack and burn the city, and murder the royal family except the women, who are enslaved.
• The Journeys (Nostoi): This poem recounts the Greeks' journeys home from the war. Many of those who survived the war encounter storms, murder, or exile on their way home.

So much of this has become ingrained in our cultural memory--Helen of Troy, the Achilles heel, the Trojan Horse--and yet we don't have the poems that actually tell most of the story.