Penelope takes Odysseus' bow down from its place of honor on the wall. She remembers how he obtained it as the payment for a debt from Iphitos from Lakedaimon.
Don't worry—Homer tells you the story: Odysseus met Iphitos in Messene, where he (Odysseus) had come claiming the natives owed Ithaka for having stolen some sheep way back when. Iphitos was also there on the account of livestock; he was tracking some stray mares that apparently wandered to Messene themselves.
But these mares ended up being the death of Iphitos, since he later wandered to the house of Herakles (Hercules), who promptly killed him so he could have the mares.
The point is, Odysseus became friends with Iphitos; he gave him a sword and spear, and Iphitos in return gave him the bow that Penelope is now taking off the wall.
Back to the Queen. She approaches the suitors and announces the contest and all its details, which we've already heard.
Eumaios and Philoitios present the weapons and both break down in tears, since they know Penelope has given up hope that her husband will ever return.
Antinoös mocks them for their sniveling, of course.
Telemachos is the first to try stringing the bow, not because he wants to marry his mother (he'll leave that to this ancient Greek hero), but because he wants to prove his strength, manliness, and virility.
After four tries, it looks like Telemachos is finally about to succeed—when beggar Odysseus signals for him not to do it. Convenient.
Telemachos obeys and hands the bow over to the first suitor, who fails miserably.
Antinoös orders Melanthios to build a fire and bring a cake of lard so that they can limber up the bow in the hopes of stringing it. (Cheating!)
As he does, beggar Odysseus notices Eumaios and Philoitios leaving the hall. He rushes after them and reveals himself as Odysseus. As proof, he shows them his scar. Woohoo!
In the meantime, Eurymachos has been shamed by the bow; he can't string it, either.
To delay his own attempt, Antinoös distracts everyone's attention with the feast and says he'll try the bow tomorrow after they have eaten.
Beggar Odysseus speaks up; he wants a chance at stringing the bow.
The suitors, especially Antinoös, emphatically say no. They're afraid he can actually do it, since they saw his absolutely awesome body a few days before.
Penelope scolds Antinoös and Eurymachos for treating the beggar so badly and invites him to give it a shot.
Telemachos uncharacteristically steps forward and tells his mother that this is a man's affair and she ought to go upstairs and be a woman. Alone. In the bedroom.