In Greek mythology, Zeus is king of the gods, and in that capacity, it was believed that he ultimately upheld the cosmic and natural order. We see this role (as supreme cosmic arbiter) reflected consistently throughout The Odyssey.

For example, consider the beginning of the poem. When Athena wishes to see Odysseus released from his captivity by Calypso, it is to Zeus she appeals, and it is Zeus who sends Hermes to secure Odysseus's release. Later, when Odysseus begins his recounting of his sufferings to the Phaecians, it is notable that he credits Zeus's primary responsibility in shaping his difficulties, stating:

Let me tell you about the voyage fraught with hardship Zeus inflicted on me, homeward bound from Troy.

That responsibility is reflected throughout the poem itself. For example, when Odysseus's men slaughter the cattle of Helios, it is to Zeus that Helios brings his complaint, and it is Zeus who throws the thunderbolt which destroyed Odysseus's ship—an attack from which Odysseus himself is the only survivor. Later, after the Phaecians have returned Odysseus to Ithaca, Poseidon himself will appeal to Zeus, and it is on Zeus's judgment that Poseidon destroys the Phaecian ship while it is in sight of Phaecia itself.

Throughout the poem, Zeus is continually invoked as the supreme authority among the gods, a role in keeping with his cosmic significance within this ancient mythology.

Although Zeus, the head Greek god, does not often appear in The Odyssey, he plays a major role in the epic. At the beginning of The Odyssey, the goddess Athena appeals to Zeus to help Odysseus, a Trojan War hero who has been stranded away from his home in Ithaca for several years. Athena wants to rescue the clever Odysseus, and she begs Zeus to help him. Zeus knows that Poseidon, the god of the sea, is angry at Odysseus because Odysseus blinded his son, Polyphemus the Cyclops. Zeus allows Athena to intervene, and Zeus promises to help Odysseus get home. Zeus says that if all the gods are on Odysseus's side, Poseidon can't stand in their way. At that point, Athena flies away to Ithaca to encourage Telemachus, Odysseus's son, to start challenging the suitors. She also says that Hermes will fly to the nymph Calypso to tell her to release Odysseus. Zeus's decision in favor of Odysseus starts the action of the epic.

Zeus also appears at the end of the epic, when Athena asks him what his plans are after Odysseus and Telemachus have slain the suitors. Zeus responds that the parties in Ithaca must come to a peace agreement and agree to making Odysseus king. They must forget their battles and work together. Therefore, Zeus's decrees begin and end the epic.

As the head of the Olympian gods, Zeus usually plays a key role in all the stories involving multiple gods (and often times in his own stories because he liked to father children). He claims that he holds Odysseus in high esteem (Bk. 1, line 63), but his brother Poseidon does not because Odysseus blinded Poseidon's son, Polyphemus.

One big role of his in the Odyssey is his support for his daughter, Athena, and her desire to help Odysseus. He allows her to go to Ithaca to tell Telemachus that his father, Odysseus, will return home (Bk. 1, line 81). He also sent two eagles as a sign to help Telemachus (Bk. 2, line 146). As is stated above, he also makes Calypso release Odysseus and his men (Bk. 5, line 29).

In this book, Zeus' role is a little lesser as compared to his role in other things, but he is still important because he is what keeps things moving along.

This is a good question. Unlike modern books, the Greeks viewed the gods as active agents in society. So, when we come to Homer's Odyssey, we see that Zeus has an important role. 

For example, Zeus is the king of the gods and he is the one who reassures Athene, a goddess, that Odysseus will eventually make it home, and that he would make it home with great riches. Here is a quote:

"They will honour him from the heart, as if he were divine, and send him on board ship to his native land, gifting him piles of gold, and bronze and garments, more than he could ever have won from Troy if he had reached home safely with his fair share of the spoils." 

Another example of the importance of Zeus is that he releases Odysseus from the island of Calypso. Calypso holds Odysseus as a prisoner. Zeus sends Hermes to tell Calypso to release him. This starts his long journey home.