Like all good epics, The Odyssey begins in media res, or "in the middle." The narrative of the poem begins in Ithaca, where Odysseus' son and wife, Telemachus and Penelope, respectively, are besieged by coarse suitors vying for Penelope's hand in marriage. Urged on by the goddess Athena, Telemachus sets off to look for his father Odysseus (who fought in the Trojan War), while Penelope keeps the suitors at bay. Meanwhile, Odysseus is stranded on Calypso's island, but he yearns to return home to Ithaca. He soon gets his wish and is able to escape, although he's shipwrecked on the Phaeacians' island. There, he recounts his previous adventures.
During Odysseus' extensive narrative, we learn of the extent of his travels, which include his encounters with Polyphemus (a hideous Cyclops), the sorceress Circe (who turned half of Odysseus' men in animals), and the frightful Scylla and Charybdis, along with a trip to the world of the dead. Following his story, we return to the present, where the Phaeacians approve of Odysseus' tale and take him back to Ithaca at last. Back on his home turf, Odysseus teams up with Telemachus (who has conveniently also returned to the island), and drives the suitors from his palace.
Any summary of The Odyssey, however, is doomed to fall far short of the actual poem. The classic piece is truly epic, as it spans many years and chronicles a quintessential human desire: the quest to find/return home. As such, while you may be able to understand the rough outline of the story with the summary, you haven't truly experienced the majesty of The Odyssey until you've read it cover to cover.