1. Circe is important because when she turned Eurylochus' exploring party into pigs, Odysseus had to go save them, which reveals some more of the good leadership qualities he has. It shows his amazing courage because he had to go down the Land of the Dead by himself.
2. Odysseus' encounter with Calypso was important because it shows him using his metis to talk her into letting him go. It also shows that near the end of his journey, the fearless leader was gone and was replaced with a desperate husband trying to get back to his loyal wife. Odysseus even cries in this section, which he never would have done before. The transformative journey shows its effect on him.
3. Polyphemus the Cyclops is important because this is the first part where Odysseus really shows his good and bad qualities as a leader. He showed good qualities when he figured out a plan to let them escape.
But then he showed his bad qualities when he let his anger and arrogance get the best of him and started taunting Polyphemus. This was very dangerous because it could have gotten him and his crew killed, but he wouldn't listen to anyone and kept on doing it.
4. Eumaeus was important to Odysseus advancing in his journey because when Odysseus came back, he was one of his only friends. Eumaeus remained loyal to Odysseus after all those years that he was gone. He helped defeat the suitors and restore the social order in Ithaca.
5. Tiresias the blind prophet of Thebes was important because he told Odysseus what was going to happen so that he and his crew would be prepared. Tiresias said that Poseidon was angry at Odysseus, that the crew would eat Helios' cattle and only Odysseus...