Genre
Homeric Epic Poem
Setting and Context
Ancient Greece, in the wake of the Trojan War
Narrator and Point of View
Homer, or whatever Bard is performing the poem at a given time, functions as a third-person omniscient narrator. Ontologically speaking, the Bard's facility as a narrator is derived from the divine blessing of the Muses, who speak through him after being invoked at the start of the epic.
Tone and Mood
The tone and mood are primarily serious and epic (no pun intended) in nature. The narrator is first and foremost concerned with relating events and character thoughts as they happened, and the poem therefore has a narrative air evoking as much transparency as is possible, given the supernatural nature of events constituting the narrative.
Protagonist and Antagonist
Odysseus is the protagonist; Poseidon is the principle antagonist, though many characters and forces frustrate Odysseus over the course of his journey.
Major Conflict
Poseidon, enraged at Odysseus blinding his son, the cyclops Polyphemus, goes out of his way to prevent Odysseus from returning home to Ithaca after the Trojan War. Meanwhile, suitors vie for the hand of Odysseus' wife, Penelope.
Climax
The climax occurs when Odysseus reveals himself to the suitors by stringing his bow, and he joins with Telemachus in slaughtering them.
Foreshadowing
Because the epic is told as a history and begins in medias res, the entire tenor of the narrative is influenced by elements of fairly overt foreshadowing.
Understatement
Given the epic nature of the poem, very little is understated. One notable exception is Achilles, particularly in relation to how he was depicted in The Iliad, the immediate predecessor to The Odyssey: once a larger-than-life hero, Achilles appears relatively ordinary when Odysseus encounters him on his visit to the underworld in Book 11; Achilles bemoans his state, saying that he would rather be a slave in the world of the living than rule amongst the dead.
Allusions
As a sequel to The Iliad, which described the events of the Trojan War, the epic is riddled with references and allusions to the fall of Troy -- up to and including Achilles' position in the underworld. There is also frequent allusion to Greek mythology and history, with which the intended audience of the epic would have been familiar -- for example, the reference in Book 12 (lines 69-72) to the Argo, the ship of Jason and the Argonauts in the myth of the Golden Fleece.
Imagery
The basic form of narrative in The Odyssey is very strongly oriented towards imagery. Consider a representative example, from when Odysseus praises Alcinous' bard: There's nothing better than when deep joy holds sway throughout the realm and banqueters up and down the palace sit in ranks, enthralled to hear the bard, and before them all, the tables heaped with bread and meats, and drawing wine from a mixing-bowl the steward makes his rounds and keeps the winecups flowing" (9.5-10).
Paradox
The most famous paradox in The Odyssey is the trick by which Odysseus outwits and escapes the Cyclops in Book 9: Odysseus tells the Cyclops that his name is "Nobody," such that when he blinds the Cyclops and the Cyclops asks for help from his countrymen, the Cyclops can only exclaim that "Nobody" is attacking him, leading his countrymen to assume that he is not in fact coming to any harm. This clever paradox turns on Odysseus' incisive choice to say that his name is a word that itself denies the existence of its referent -- a paradoxical turn closely related to the philosophical problem of negative existentials (see bibliography for reference).
Parallelism
Though there are many differences between the two, Telemachus and Odysseus both undergo odysseys of sorts over the course of the epic: as Odysseus is physically alienated from his home, so too is Telemachus alienated from his right over his home by the suitors who has usurped control of it. The parallel paths of father and son converge when the two slaughter the suitors and reclaim their home through nostos -- i.e., homecoming.
Metonymy and Synecdoche
Neither is particularly prominent -- the figurative language of the epic is more typically characterized by metaphor, simile, and personification.
Personification
The most blatant instance of personification in the poem is in Book 10, when Circe uses magic to transform Odysseus' men into pigs. These are personified pigs in the most literal sense, because, as Homer says, "only the men's minds stayed steadfast as before" (264-265). The sunrise is also personified as 'rosy-fingered Dawn'.