First, the Sparknotes summary:
Penelope gets Odysseus’s bow out of the storeroom and announces that she will marry the suitor who can string it and then shoot an arrow through a line of twelve axes. Telemachus sets up the axes and then tries his own hand at the bow, but fails in his attempt to string it. The suitors warm and grease the bow to make it supple, but one by one they all try and fail.
Meanwhile, Odysseus follows Eumaeus and Philoetius outside. He assures himself of their loyalty and then reveals his identity to them by means of the scar on his foot. He promises to treat them as Telemachus’s brothers if they fight by his side against the suitors.
When Odysseus returns, Eurymachus has the bow. He feels disgraced that he cannot string it, because he knows that this failure proves his inferiority to Odysseus. Antinous suggests that they adjourn until the next day, when they can sacrifice to Apollo, the archer god, before trying again. Odysseus, still disguised, then asks for the bow. All of the suitors complain, fearing that he will succeed. Antinous ridicules Odysseus, saying that the wine has gone to his head and that he will bring disaster upon himself, just like the legendary drunken centaur Eurytion. Telemachus takes control and orders Eumaeus to give Odysseus the bow. Needless to say, Odysseus easily strings it and sends the first arrow he grabs whistling through all twelve axes.
One thing which this summary fails to make clear is that Telemakhos doesn't fail to string the bow - he comes very close to stringing it, but is silently warned by Odysseus to give it up and then pretends that he isn't strong enough. This is one of several indications in this Book - including a rebuke which he delivers to Penelope - which indicate again that Telemakhos is stepping into Odysseus' role as master of the house. Telemakhos' speech to Penelope finishes like this:
....The bow shall be the concern of all the
men, and me especially; for mine is the power in the house.
Also worth quoting are the final, ominous lines of the Book, in which father and son stand together before the suitors:
He spoke and signalled with his eyebrows; Telemakhos strapped
on his sharp sword - the dear son of godlike Odysseus -
and grasped his spear with his hand, and stood beside
Odysseus, by his chair, armed with shining bronze.