Odysseus and his fleet arrived on the island of the Cyclops after sailing from the land of the Lotus-eaters. They already had plenty of wine aboard their ships after having sacked the city of the Citons, and killing a number of goats that ran wild on the island, they feasted "the livelong day." The next morning, Odysseus led a party of his men into a large cave; soon the Cyclops arrived and Odysseus made their presence known. Despite Odysseus' mention of Zeus and Poseidon, the Cyclops showed no fear of the gods. Suddenly, the giant grabbed two of Odysseus' men and bashed them on the ground, splattering their brains before tearing "them limb from limb." Then he ate the bodies--"bones, marrow and entrails."
The specific Cyclops that they face is named Polyphemus, whose father happens to be the sea god Poseidon. Cyclopses are a race of giant one-eyed monsters that live on this island.
The important thing about the Cyclops in the context of ancient Greece is that they are uncivilized. Being "civilized" was an incredibly important virtue to the Greeks. (Although they thought ANY non-Greek was uncivilized.) Of course it is a monstrous and villainous thing to do to grab two men, kill them, and eat their bodies--but what is notable about this is the fact that right before this happens, Odysseus tries to invoke xenos. Xenos is the Greek virtue of hospitality, which is tied with civilization. Homer's audience would have recognized this immediately. Polyphemus ignores the invocation to xenos, and does not act like a gracious "host" would.
Later on, Polyphemus also drinks some unwatered wine that Odysseus gives him. In ancient Greece, this was seen as the ultimate sign of barbarism. Civilized Greeks only drank wine that was diluted with water. To them, only uncivilzed barbarians drank undiluted wine.