How does the epic end?
The poem ends unexpectedly, in my opinion. Homer finishes off the story with the suitors' appearance in the underworld, where Agamemnon makes a final appearance, rounding out the parallels between the Odysseus story and the story of the fall of the House of Atreus.
Then, after his reunion with Penelope, Odysseus makes one final journey - but not to fulfill the command of Tireseus, and appease Poseidon. Instead, he goes to pay a visit to his father. The relatives of the dead suitors seek their revenge, and it appears that there will be another extended blood-letting, but Athena intervenes and puts an end to the fighting.
The important question to ask is not how does the poem end, but why does it end as it does. Homer doesn't provide any direct answer; that task is left to you, the scholar.
Hmm. How much of the ending do you want? The key elements are that Odysseus comes home (in disguise, at first), that the suitors and killed and their souls guided to the underworld, and that Odysseus, having been away so long, has to prove to Penelope that he is who he claims he is.
There are lots of wonderful specific details along the way, though.
If you mean the very end of the poem (the final book), Odysseus is reunited with his aged father and, after some violence, they unify the divided political factions on the island.