Homer's Odyssey tells the story of its protagonist Odysseus and his travels through many strange lands before coming home to Ithaca at last. The epic begins ten years after the events of the Iliad (the fall of Troy). Odysseus still hasn't returned home.
Much of Greece is surrounded by water and it is unsurprising, then, that Odysseus stops on many islands before returning to Ithaca. Indeed, Ithaca itself is an island. He is trapped on many of the islands he lands on because the islands are often inhabited by terrible creatures.
The first we hear of an island is in Book IX when Odysseus is telling his tale to Phaeacians. He tells them of how his ship was blown off course and how he ended up in the land of the Lotus-Eaters and the island of the one-eyed cyclops Polyphemus. He escapes the cave of Polyphemus with the aid of a linguistic trick (calling himself 'Noman' or 'Outis' in Greek).
Then, they end up on the floating island of the god Aeolus, who at their departure gives Odysseus a leather bag containing all the winds except the favorable West Wind.
The third island where they are trapped is in Book X of the Odyssey: the island of Circe. Circe is a sorceress who turns Odysseus's crew into pigs. Odysseus senses danger and stays behind, avoiding their fate. With the help of Hermes, he manages to resists Circe's enchantments and succeeds in freeing his crew from Circe.
In Book XII, after passing Scylla and Charbydis, they come to the island where the sacred cattle of the Sun are to be found. Although they have been warned against harming them, they are stranded there and hunger leads them to slaughter and eat some upon which they are punished by the Zeus (to whom the Sun complains).
With the ship destroyed, Odysseus alone drifts to Calypso's island, clinging to the mast, which is all that remains. After seven years with Calypso, he finally builds a raft and leaves her island.