The exposition (or introduction) of Homer's The Odyssey introduces the character of Odysseus being discussed by Athena and Zeus, her father. She wants to know why Odysseus has not been able to return home, even though the Trojan War has been over for ten years.
Then Athena said, “Father, son of Cronus, King of kings...it is for Odysseus that my heart bleeds, when I think of his sufferings in that lonely sea-girt island, far away, poor man, from all his friends.
Later the reader learns the details of Odysseus' trials to return home as he tells of his adventures and trials (through Book 12). In this introductory part of the story, Zeus alludes to the longer story of the Cyclopes, Polyphemus—the son of Poseidon. Because Odysseus had behaved dishonorably toward Polyphemus (even though the giant ate some of Odysseus' men), Poseidon has punished Odysseus by keeping him away from home. At that moment, though, because Poseidon is not present, Zeus releases Odysseus so he may return to his wife and son. It is also in this section (Book 1) that Athena instructs Telemachus (Odysseus' son) to seek out what information he may be able to find about his father.
In general the reader learns that Odysseus was at war; that the Greek gods are involved in the story; that our hero angered one god who kept him another ten years from his home; and, that his son has not given up hope of his return.
The rising action starts with Book Two and continues throughout the following books as we learn of Odysseus' adventures, his travels towards Ithaca, the suitors who have overrun his home, the attempt on his son's life, and Odysseus' arrival in Ithaca. We see him reunited with two of his servants and his son, and learn of the suitors' treatment of the disguised Odysseus. Odysseus has plans to punish the suitors, and Penelope has placed a test of strength before them to see who (so she says) she will marry—based on who can complete the test. (No one but Odysseus can do so.)
The climax of the story takes place when Odysseus, Telemachus and two servants attack and kill all the suitors. At one point Odysseus calls on Athena, wondering how he will be able to bring the suitors to their just punishment. Athena chides and encourages him, saying:
Am I not a goddess, and have I not protected you throughout in all your troubles? I tell you plainly that even though there were fifty bands of men surrounding us and eager to kill us, you should take all their sheep and cattle, and drive them away with you.
The falling action includes Odysseus' reunion with Penelope: how she cannot believe it is him and how she tests him to make sure he is who he says he is—She tells her servant to move Odysseus' bed back into his room; only Odysseus would know this to be impossible as he built the bed from a tree, and constructed the room around the bed! She says:
Euryclea, take his bed outside the bed chamber that he himself built. Bring the bed outside this room, and put bedding upon it with fleeces, good coverlets, and blankets.
When Odysseus grows angry, wondering who could have moved the bed, Penelope is convinced this is her husband, and she finally welcomes him home.
The resolution (conclusion) describes how the suitors arrive in Hades; how Odysseus goes to see his elderly father, Laertes; and, how the relatives of the suitors come to take revenge on Odysseus. We learn of the ensuing fight and Athena's ability to bring about peace between them all.