What happened to the men who sailed from Troy homeward with Odysseus?
Odysseus was a Greek general and king of Ithaca fighting in the Trojan War, whose name means "trouble" in Greek and is known to be both quite intelligent as well as possessing the tragic flaw of hubris. As part of his trip back to Greece, Odysseus and his crew encountered many difficulties, extending their voyage to a total of 10 years in length. Chief among the encounters was that of Polyphemus, as its consequences became a mark for the travails of the rest of the journey.
Odysseus' encounter with the cyclops following his landing on Sicily. Polyphemus ends up eating several of the crew and traps the rest with a large boulder over his cave entrance while he grazes his sheep. Odysseus tricks the giant by getting him drunk, blinding him with a spear and then hiding under the sheep with his men as they are let out to graze, fooling the cyclops. Unfortunately, Odysseus made the mistake of revealing his real name while barely avoiding the huge rocks Polyphemus threw at his ships as they left. Now knowing the name, the giant prayed to his father, Poseidon, for revenge. Thus, Poseidon made sure Odysseus' voyage home would not be a brief one...
Other noteworthy excursions during the journey back to Ithaca are Circe the witch, in which he and his men are turned into pigs. Charybdis and Scylla are two narrow inlet sea monsters that Odysseus must decide to sacrifice some of his men to Scylla, or they would be unable to sail past either. The Island of the Lotus Eaters also represents the character's intelligence, as eating of the lotus causes a perpetual sleep and overwhelming apathy for life and goals that he only overcomes through great force of determination that would likely doom a lesser man. These two events perhaps best demonstrate Odysseus as a character willing to make difficult decisions, rather than just a heroic man of great strength who is able to overcome all challenges, akin to Heracles.
During these delays, his wife Penelope had to constantly deal with aggressive suitors assuming that the king was dead. She cunningly delayed them by claiming to be weaving a funeral shroud that she then proceeding to unravel its progress at night, delaying them until the deception was revealed. She is often seen as a symbol of fidelity in marriage, but recent feminist interpretations have different views. While there are a great many more difficulties during the Odyssey such as the sweet sounds of the Sirens, the above is a sufficient start for this answer without necessarily summarizing the whole poem.