Odysseus is angry with Penelope because she has allowed so many suitors to occupy his household in his absence while he is presumed dead. Penelope is considered excitable and a little flighty, so she's left in the dark about most of the goings on in the story. However, she has not completely given up hope that he may not be dead so she constantly cleverly puts off her suitors by promising that she will marry one of them when she finishes Odysseus' burial shroud (though she does not actually plan on finishing it) and it can be argued that she knew that Odysseus was back before he let her know because she agrees to marry the best archer, and she knows Odysseus is the best archer.
Odysseus then kills all of the suitors and he and Penelope presumably live happily ever after.
Near the end of the poem, after Odysseus has returned to Ithaca and slain Penelope's suitors, he chastises her for her unwillingness to believe that he is who he says. He calls her "stubborn" and asks a servant to make a bed for him elsewhere so that he can sleep alone. In order to test his honesty, Penelope insists that the servant move the frame of his bed from their bedroom so that he can sleep in his own bed. However, the bed that Odysseus crafted for them many years ago would be nearly impossible to move since it was carved into and from an olive tree that grows up through the room. When Odysseus becomes enraged, thinking that she has allowed someone to destroy that bed, he describes it perfectly, and this lets Penelope know that he is, in fact, her husband. She is quite cunning and intelligent, and she quickly devised a perfect test to ascertain the truth. Once Odysseus stops talking, she throws "her arms round Odysseus' neck and kissed his face." She explains that she worried that some man could come and trick her with his story about being Odysseus, and so she had to be sure it was him. This resolves their conflict and they basically stay up all night to talk.