As the name suggests, a product owner should own the product on behalf of the company. The individual is responsible for ensuring that the product creates value for its customers and users as well as the company providing it. You can think of the product owner as the person who champions the product, who facilitates the product decisions, and who has the final say about the product, for instance, if and how feedback is actioned and which features are released.
The following diagram provides a summary of how I view the role of the product owner for commercial products and it is based on my product management framework.
As the picture above shows, a product owner should have strategic product management skills, such as product strategy and roadmapping, as well as tactical ones, including product backlog management and user stories. I have circled the areas, which are required by Scrum—the framework in which the role originated. The other areas are necessary in my experience to allow the product owner to do a great job and achieve product success even though they are not mandated by Scrum (or other agile models).
If you work as a product owner for in-house applications, you should adjust the picture above: consider replacing "Marketing" with "Operations" and removing "Sales and Support". Similarly, if you manage a tech product, like a physics engine, you may want to promote the "Development/Technologies" area and move it to the inner circle, as in-depth technical knowledge is usually required to describe the products and its APIs.
As the product owner, you should directly interact with the customers and users, the development team, and other key stakeholders, as the picture below shows.
I have circled the Scrum team, the unit consisting of product owner, ScrumMaster and development team in the picture above to indicate that the product owner should have a close and trustful relationship with the other Scrum team members: Scrum views the product owner as part of the wider Scrum team. This makes sense, as great products emerge when the product owner takes her market and business knowledge and collaborates with the development team.
You can learn more about the role of the product owner by:
The Scrum Product Owner Role on One Page
The product owner is a key role in Scrum. Read this post from expert Roman Pichler for a crisp summary of the scrum product owner role with its main responsibilities.