First, the Sparknotes summary:
When the suitors retire for the night, Telemachus and Odysseus remove the arms as planned. Athena lights the room for them so that they can see as they work. Telemachus tells Eurycleia that they are storing the arms to keep them from being damaged.
After they have safely disposed of the arms, Telemachus retires and Odysseus is joined by Penelope. She has come from the women’s quarters to question her curious visitor. She knows that he has claimed to have met Odysseus, and she tests his honesty by asking him to describe her husband. Odysseus describes the Greek hero—himself, capturing each detail so perfectly that it reduces Penelope to tears. He then tells the story of how he met Odysseus and eventually came to Ithaca. In many respects, this story parallels those that he told to Athena and Eumaeus in Books 13 and 14, respectively, though it is identical to neither. He tells Penelope that, essentially, Odysseus had a long ordeal but is alive and freely traveling the seas, and predicts that Odysseus will be back within the month.
Penelope offers the beggar a bed to sleep in, but he is used to the floor, he says, and declines. Only reluctantly does he allow Eurycleia to wash his feet. As she is putting them in a basin of water, she notices a scar on one of his feet. She immediately recognizes it as the scar that Odysseus received when he went boar hunting with his grandfather Autolycus. She throws her arms around Odysseus, but he silences her while Athena keeps Penelope distracted so that Odysseus’s secret will not be carried any further. The faithful Eurycleia recovers herself and promises to keep his secret.
Before she retires, Penelope describes to Odysseus a dream that she has had in which an eagle swoops down upon her twenty pet geese and kills them all; it then perches on her roof and, in a human voice, says that he is her husband who has just put her lovers to death. Penelope declares that she has no idea what this dream means. Rising to the challenge, Odysseus explains it to her. But Penelope decides that she is going to choose a new husband nevertheless: she will marry the first man who can shoot an arrow through the holes of twelve axes set in a line.
Time is now really running out for the suitors. We live in a world of gods, omens and premonitions, and it looks like Penelope has received a pretty clear supernatural promise that she is about to be reunited with Odysseus and that Odysseus is going to kill the suitors.
Odysseus and Penelope share some poignant exchanges in this Book, with Odysseus telling some more of his "Cretan lies", claiming again that he is a higher-born man who is down on his luck. Penelope does not yet realise whom she is talking to. Eurykleia, by contrast, recognises Odysseus from an old hunting wound, and he has to forcibly shut her up. She refers to him as "teknon emon", "my son". The story of the wound allows Homer to go off into a digression about Odysseus' youth.
As in Book 18, we continue to be told that Telemakhos has now reached manhood and is now able to assume the role of master of the house.