In the ninth book of Homer's Odyssey, Odysseus tells the Cyclops Polyphemus that his name is "Nobody." Later, when Odysseus asks the Cyclops what gift of guest-friendship the Cyclops will give to him, the Cyclops tells Odysseus that the gift will be that he will eat Nobody last.
One of the definitions of irony, with respect to literature, is that the audience of a work understands something that the characters in the literary work do not understand: this is called dramatic irony. Thus, the Cyclops means one thing by his words, but the audience knows that things will actually turn out differently.
So, Homer's audience knows something the Cyclops does not know: namely, that Odysseus' real name is not Nobody. Homer's audience is also smarter than the Cyclops because they understand that eating Nobody means that the Cyclops will not eat anyone.
Perhaps a further irony here is that once Odysseus reveals his real name to the Cyclops (after Odysseus has blinded the Cyclops), this leads to the Cyclops praying to his father Poseidon to take vengeance against Odysseus. Poseidon hears his son's prayer and causes Odysseus many sorrows upon the open sea.
There is also some wordplay going on as a result of Odysseus telling Polyphemus that his name is actually "Nobody." Unbeknownst to the Cyclops (hence, the dramatic irony identified by the other commenter), this isn't Odysseus's real name, and so his assertion that he will eat Nobody last could be considered ironic; he means he will eat Odysseus last, but he actually stumbles on the truth: soon he will eat no more of Odysseus's men.
Moreover, Polyphemus tells Odysseus that his guest-gift will be that the Cyclops will eat him last. However, this is hardly a gift! If Polyphemus eats Odysseus last, then Odysseus will have to watch the monster devour each of his other men, prolonging his sorrow (and perhaps guilt) over their loss of life. Since this statement, too, defies expectation by suggesting that it would be a gift to prolong Odysseus's life (which would lead him to suffer more), it could be considered ironic.