In a very broad sense, the conceit of The Odyssey is itself ironic: ordinarily, we think of journeys primarily entailing leaving home and going elsewhere, yet this story is entirely concerned with the hero's effort to return home.
There is a sort of painful irony associated with the fact that Achilles, who lived with the purpose of dying a glorious death, now only mourns the loss of his life, in the underworld.
The deception of the Cyclops
Odysseus' defeat of the Cyclops by paradoxically naming himself "Nobody" is ironic because the very name which Odysseus used prevents him from being named as an aggressor against the Cyclops.
The humanity of the gods
Though we might expect gods to be all-powerful, all-knowing, and perfectly beneficent, The Odyssey illustrates a Pantheon that is instead largely characterized by petty, human-like squabbles.
The Odyssey Questions and Answers
The Question and Answer section for The Odyssey is a great
resource to ask questions, find answers, and discuss the novel.