The Product Owner's primary goal is to ensure the vision that was asked for by the stakeholders/customers is being executed by the team. The Product Owner does this by working with the stakeholders to understand the functionality of the system under development. From there, the Product Owner will write user stories or work jointly with the customers to write them. The stories go into the Product Backlog.
The Product Owner manages and represents the interests of the stakeholders and customers.
Think about the car model from the ScrumMaster section. If the ScrumMaster is like the oil and sensors in the engine, the Product Owner is the driver. The product owner points the car in the correct direction and makes the minute course adjustments necessary to stay on course and deliver results. She represents customer and stakeholder interests. Her job is to establish, nurture and communicate the product vision to the team and the other stakeholders, manage the project return on investment and project financials and make decisions about when to create an official release (with customer and stakeholder input). The Product Owner is the one person ultimately responsible for the success or failure of the project. She decides what is developed, when it is developed, and whether that which has been developed meets expectations.