The Greeks and Romans always thought that the legendary poet Homer was the greatest author ever. However, we know hardly anything about the man or men who composed the Iliad (a long epic on the wrath of the warrior Achilles, which endangered the Greek expedition to Troy) or the Odyssey (an equally long poem on the difficult voyage home of Odysseus). At the moment, most scholars agree that these texts were dictated to a writer by a very capable bard, who used older, oral traditions, at the beginning of the eighth century BCE. We do not know to what extent later poets have made additions or changes to the two epics.
It is interesting to compare the two poems of Homer with the standard version Babylonian Epic of Gilgameš. There's a similarity: a capable poet writes down a poem that had until then been transmitted orally. The main hero of the Iliad, Achilles, is a great warrior, but his greatest victory is not won by killing enemies, but by triumphing over his wrath; similarly, Gilgameš triumphs over his longing for immortalty and learns to accept that he is only human.
There's also a difference. We can not say something meaningful about the contents of the pre-Homeric poetry, and it is very difficult to say whether a Trojan War has ever taken place. The "prehistory" of the Epic of Gilgameš, on the other hand, is pretty well-known.
Many Greek and Latin authors were consciously influenced by Homer's language, and several people tried to emulate the Homeric heroes (e.g., Agesilaus II and Alexander the Great). In Egypt, he received divine honors.
His most important influence, however, must be sought somewhere else. Unlike contemporary sources from other cultures, Homer's poems are more or less "objective". An Egyptian text leaves no doubt that the enemy of pharaoh are evil impersonated. Homer, on the other hand, offers a balanced judgment of the Trojans and Greeks. This objectivity is not unique in the ancient world - Babylonian chronicles have no difficulty in admitting defeats of Babylonian rulers - but it is rare in ancient literature. Through the Histories by Herodotus of Halicarnassus, this may be the Poet's greatest legacy to western civilization.
Several other poems were attributed to Homer, some of them belonging to the Epic Cycle. The Hymns have survived.