How is heroism portrayed in The Odyssey?

In The Odyssey, heroism is presented in the traditional ancient Greek manner, as consisting in acts of macho bravado imposed by male warriors upon the world. There's nothing remotely humble or quiet about the heroism on display here; being a hero involves big and bold statements that let everyone—both mortals and immortals alike—know who you are and what you're capable of.
That's not to say that heroes are depicted as all brawn and no brain; Odysseus certainly doesn't fall into that category. Whether it's having himself tied to the mast while his ship passes the Sirens or devising a cunning plan to help him and his men escape the clutches of the deadly Cyclops, Polyphemus, Odysseus's heroism involves a great deal of thought and planning.
Yet even here, the overriding dangers of hubris are never very far away. After escaping from Polyphemus, Odysseus openly taunts the Cyclops he's just blinded, revealing his identity as he sails away from the island. What this episode illustrates is that there's often a fine line between heroism and arrogance, between courage and impetuosity. Odysseus hasn't learned this valuable lesson just yet, but thanks to his big mouth he soon will—and the hard way, too.

In The Odyssey, heroism is portrayed as something which men are capable of -- Penelope is cunning and clever, but she is never discussed as heroic -- and heroism is achievable by fighting external enemies (like monsters), overcoming physical challenges (like going to the Underworld) with bravery, and exhibiting good character.  Odysseus does all these things.  He goes up against monsters like Scylla and Charybdis and Polyphemus, but he also exhibits good character qualities when he refuses to hide below deck, for example, when his ship must pass Scylla.  He knows that she will likely eat six of his men, and he casts his lot with the rest of them, standing on deck and brandishing his sword against her.  Further, when he goes to blind Polyphemus so that he and his men can make their escape, he takes it upon himself to lead the most dangerous part of that mission: actually stabbing the Cyclops in the eye.  Heroism is thus presented as something that a man can achieve by taking on the evil or the monstrous as well as by putting the needs of others before his own.