Campbell's idea of the "hero's journey" consists of three basic movements: separation or departure, initiation, and return. Each movement is subdivided into smaller elements (there are thirteen of these; I won't list them all, but a quick search on "monomyth" should bring them up for you).
The Odyssey can be mapped to these movements. Odysseus's departure, of course, occurs long before the events in The Odyssey, with Agamemnon's call for him to join the Greek forces in the Trojan War. This "call to adventure" starts Odysseus on a journey of personal growth in which he must overcome many obstacles.
The initiation phase includes the many trials Odysseus must overcome in his journey, and consists of many of the episodes for which the poem is famous. Each of these trials teaches Odysseus a lesson. For instance, when Odysseus defeats the cyclops, he learns that he must outthink his opponent rather than relying on strength, and also must develop humility. In the case of the sirens, he learns that he needs the help of others to overcome some obstacles.
The return stage comprises Odysseus's return to Ithaca and his home. Odysseus returns a changed man, and the lessons he has learned on his journey that enable him to appreciate his home and his wife, and defeat the suitors.