First, the Sparknotes summary:
At Pylos, Telemachus and Mentor (Athena in disguise) witness an impressive religious ceremony in which dozens of bulls are sacrificed to Poseidon, the god of the sea. Although Telemachus has little experience with public speaking, Mentor gives him the encouragement that he needs to approach Nestor, the city’s king, and ask him about Odysseus. Nestor, however, has no information about the Greek hero. He recounts that after the fall of Troy a falling-out occurred between Agamemnon and Menelaus, the two Greek brothers who had led the expedition. Menelaus set sail for Greece immediately, while Agamemnon decided to wait a day and continue sacrificing on the shores of Troy. Nestor went with Menelaus, while Odysseus stayed with Agamemnon, and he has heard no news of Odysseus. He says that he can only pray that Athena will show Telemachus the kindness that she showed Odysseus. He adds that he has heard that suitors have taken over the prince’s house in Ithaca and that he hopes that Telemachus will achieve the renown in defense of his father that Orestes, son of Agamemnon, won in defense of his father.
Telemachus then asks Nestor about Agamemnon’s fate. Nestor explains that Agamemnon returned from Troy to find that Aegisthus, a base coward who remained behind while the Greeks fought in Troy, had seduced and married his wife, Clytemnestra. With her approval, Aegisthus murdered Agamemnon. He would have then taken over Agamemnon’s kingdom had not Orestes, who was in exile in Athens, returned and killed Aegisthus and Clytemnestra. Nestor holds the courage of Orestes up as an example for Telemachus. He sends his own son Pisistratus along to accompany Telemachus to Sparta, and the two set out by land the next day. Athena, who reveals her divinity by shedding the form of Mentor and changing into an eagle before the entire court of Pylos, stays behind to protect Telemachus’s ship and its crew.
Telemakhos has now travelled from the island of Ithaka in north-west Greece to the town of Pylos in south-west Greece, in search of news of Odysseus. The Book begins and ends with the communal piety of public sacrifices, a sign that the characters involved (unlike, say, the suitors on Ithaka) are maintaining a proper relationship with the gods. Athene stays in the action, too, and we have the unusual spectacle of her pretending to be a human praying to a god and secretly granting her own prayer.
Note that Nestor shows himself to be both hospitable, treating Telemakhos with consideration and generosity as his guest, and pious, praying and sacrificing to the gods.
Telemakhos is hesitant to question Nestor: he protests that he is unskilled at speaking, and anyway it is a disgrace for a youth to question an elder. In the event, however, he doesn't seem tongue-tied at all. In reply, the garrulous old man retells a story that Zeus has already alluded to in Book 1 - the story of Orestes. In short, while King Agamemnon was away fighting the Trojan War, his wife Klytaimnestra was seduced by the villain Aigisthos. When Agamemnon came back home, Aigisthos killed him. The parallels to the story of Odysseus, Penelope and the suitors don't need to be emphasised. The story has a happy ending, though, for Agamemnon's son Orestes turns up and kills both Aisigthos and his mother. Note that there is no moral censure attached to his deed (though this will change in later Greek literature). This is not a society where you call the police when things start kicking off. Blood vengeance is the way these things are handled, even if it involves knifing your mum.
The question is whether Telemakhos will have the bottle to sort out his household like Orestes did, albeit in his case no matricide will be involved. We are implicitly reassured that he will. Old Nestor marvels at how much he talks like Odysseus. Again, Homer is telling us that he is a chip off the old block.
Finally, the themes of reputation and fame (kleos) continue to appear, as they have in the previous books. These things counted for a lot in Homeric society.