A brief orientation
"Homer" is the name the Greeks gave to the author of the Iliad and the Odyssey. Whether a single historical poet by that name composed both works, whether the two were composed by different poets, and whether the texts as we have them stand at the end of a long oral tradition of bardic performance — these are the questions of what modern scholarship calls the "Homeric Question," and they are open.
The conventional dating places the composition of the poems in roughly the eighth century BCE, with substantial textual stabilisation in the sixth and again in the work of the Alexandrian scholars (Zenodotus, Aristophanes of Byzantium, Aristarchus) in the third and second.
What the poems are
The Iliad takes a small slice of the tenth and final year of the Greek war against Troy and turns the wrath of Achilles into the subject of an epic of around 15,000 lines, divided in transmission into twenty-four books. The Odyssey tells the homecoming of Odysseus over the years following the war and is similarly divided into twenty-four books. Both poems are in dactylic hexameter; both were the foundation of Greek paideia for as long as there was a Greek paideia to speak of.
Why he matters for Virtue & Power
The classical philosophical tradition is unintelligible without Homer in the background. Plato's Republic spends considerable space arguing with the place of Homer in education; Aristotle quotes him as a moral authority; the entire ancient inquiry into courage, war and peace, honour and the shape of a heroic life begins with the poems and continues by arguing with them. Reading the philosophers without Homer is reading the second half of the conversation.