Odysseus is one of the most impressive of the Greek heroes because there aren't any glaring, enormous character flaws that jump out at us upon reading his story. Unlike Achilles, whose anger is the primary theme of the Iliad, Odysseus appears to be the "perfect guy" - brave, loyal, and really smart (probably handsome, too, since these minor goddesses all want him to stay with them forever!).
Odysseus' one weakness appears to be his pride. In Book 9, Odysseus describes his adventures with Polyphemus, the Cyclops. When escaping from the Cyclops, everything would have gone smoothly for Odysseus and his men if Odysseus had just kept his mouth shut, rather than bragging about who he was. Telling the Cyclops that he was Odysseus caused a ton of problems as that gave the Cyclops a chance to complain to his father, Poseidon, about this awful person who blinded him. Being the sea god, Poseidon worked himself into a rage and did his best to destroy Odysseus at sea, thus prolonging Odysseus' journey home to Penelope.
But again, Odysseus shows us to be a really great Greek Hero as he learns from his mistakes and tempers his pride and arrogance later in the story, thus ensuring that he can make it home, defeat the suitors, and be reunited with Penelope.
Be sure to check the eNotes links below for more information!