First, the Sparknotes summary:
Eurycleia goes upstairs to call Penelope, who has slept through the entire fight. Penelope doesn’t believe anything that Eurycleia says, and she remains in disbelief even when she comes downstairs and sees her husband with her own eyes. Telemachus rebukes her for not greeting Odysseus more lovingly after his long absence, but Odysseus has other problems to worry about. He has just killed all of the noble young men of Ithaca — their parents will surely be greatly distressed. He decides that he and his family will need to lay low at their farm for a while. In the meantime, a minstrel strikes up a happy song so that no passers-by will suspect what has taken place in the palace.
Penelope remains wary, afraid that a god is playing a trick on her. She orders Eurycleia to move her bridal bed, and Odysseus suddenly flares up at her that their bed is immovable, explaining how it is built from the trunk of an olive tree around which the house had been constructed. Hearing him recount these details, she knows that this man must be her husband. They get reacquainted and, afterward, Odysseus gives his wife a brief account of his wanderings. He also tells her about the trip that he must make to fulfill the prophecy of Tiresias in Book 11. The next day, he leaves with Telemachus for Laertes’ orchard. He gives Penelope instructions not to leave her room or receive any visitors. Athena cloaks Odysseus and Telemachus in darkness so that no one will see them as they walk through the town.
Telemakhos is pleased at how things have turned out, but Odysseus brings him back to earth by reminding him that they have just killed over 120 people, most of them powerful aristocrats. He orders Phemios to play music so that passers-by will think that a party is in progress.
Penelope plays it cool and ends up angering Odysseus so that he is provoked into revealing, from his knowledge of the construction of their bedroom, that he really is her long-lost husband. They then have a good cry and a cuddle for several hours (while Athene delays the onset of dawn), followed by some sex. Note that Penelope is the only character in the epic who ever outsmarts Odysseus.
Odysseus lets Penelope know that he has another journey to make, as mandated by Teiresias, after which he will die in comfortable old age.
The touching portrait of Odysseus and Penelope is at odds with the clichéd image of ancient Greece as a patriarchal society in which love played little part in marriage. It is important to Homer that the couple really love each other. It seems that his ideal for a married couple is the state that he elsewhere refers to as homophrosyné, "thinking the same". It is noteworthy that Penelope's feelings on being reunited with Odysseus are compared with those of a man who has survived a shipwreck - the simile, in effect, briefly makes her into Odysseus.
Eurykleia's closeness to the family is shown again. She calls Penelope "my child", "teknon emon".