Who is narrating "The Odyssey?"
The Odyssey was originally narrated aloud by a poet or a singer. Since the tale was passed on to others through an oral tradition, the "narrator" just happened to be whomever was reciting the epic poem.
But in terms of the the verbal or textual narration, this is an example of third person omniscient narration. The narrator is not a character. The narrator knows it all (omniscient) and has insights into all events and even the actions of the gods as well as the mortals. Think of the narrator as some poet with a bird's eye view of all that occurs in the story. Homer is the author but the narrator is best understood as an anonymous poet, narrating from the perspective of third person.
There is one partial exception. In Books IX through XII, Odysseus takes over the narration, but even here, his narration is within the framework of a third person narrator. Book IX begins with "And Odysseus answered" - this continues with the third person narration. It would be Odysseus as first person narrator if it began with "I answered" but this is not the case.
An added note: Aristotle divides the poetic forms into "no narrator" (drama), "one narrator" (poetry), and "more than one narrator" (epic).