She does not marry one of the suitors.
Even though Odysseus has been gone for all of his son's childhood and many have declared him dead, Penelope stays loyal to him and refuses to marry one of the suitors.
She does allow the suitors into her home as part of their social culture. She delays the suitors advances with various excuses. She says she will not choose one until she has weaving the burial shroud, but she unravels the shroud at night to make it take longer. She also puts the suitors to a test. When they learn of this, they demand she choose one immediately. She tells them she will marry whoever can shoot an arrow through the ax handles. Odysseus returns home in disguise and succeeds in shooting the arrow before revealing his true identity.