I will marry one of you if you string my lord's bow, and shoot it through 12 axe heads.
Argos was Odysseus' puppy, but since Odysseus went to war, he never went on a hunt. He got older, and was treated terribly while Odysseus was away. He recognized him, wagged his tail, then passed away.
Eumaeus, the swineherd, was talking to Odysseus while he was in his disguise. This is a very ironic moment because Eumaeus is talking about Odysseus to Odysseus.
Fighting the Suitors
Penelope has lost hope of Odysseus coming back. She truly doesn't want to marry any of the suitors, so she makes a close to impossible challenge for them: an archery contest.
Penelope & Odysseus
While still in his disguise, Odysseus strung his bow, and shot the arrow through all the axeheads.
Odysseus, his men, and later Athena are up against over 100 suitors. They battled, and then killed all of them. Also, Athena later makes Odysseus "pretty" again.
Penelope thinks Odysseus is who he says he is, and tells the servant to make him a bed outside the bedchamber. Odysseus got mad and described the details of the bed that he had made. Penelope knows for sure it's him, and they have a sweet reunion.