The responsibility of a product manager depends on their context within an organization. Here are some examples of variables that might change a product manager's responsibilities:
Management structure. In some cases a product manager has the mandate and influence to push their product forward as they see fit, in other cases they will be responsible for executing a vision for the product defined by management.
Maturity of the product. In some cases a product manager will be responsible for launching a new product, in other cases they will be responsible for evolving an established one.
Given the variability of the product manager role, here's an attempt to enumerate the primary responsibilities of a product manager in a way that is true across different environments. Note that this list is more about what a product manager is required to do than what it takes for a product manager to thrive.
Be their company's expert in the space in which the product exists. This might mean advising management that their product should be fundamentally reconsidered given the company's position in the market.
Crystalize and articulate the vision for evolving the product and its progress toward that vision. Whether or not the product manager led the development of their product's vision, it is their job to clearly communicate the vision throughout their organization and keep stakeholders up-to-date on where the product stands in relation to that vision, its successes and failures.
Be the translator and diplomat unifying all people in the company vested in the product. Different people within an organization, with their own unique goals, will have different desires for how the product will develop. Sometimes various people will have trouble communicating with each other (e.g., sales and engineering). The job of the product manager is to be the glue that connects and unifies all interested parties.
Represent the customer. The product manager must be the company's expert on how people interact with their product (e.g., through user testing, surveys, analytics) and apply this knowledge in defining the product roadmap.
Define and represent the product roadmap. The product manager owns the master list of possible changes to the product, and is responsible for representing the rationale behind prioritization.
Own the framework of quantification. The product manager is in charge of figuring out how to measure the product's progress towards company goals and translate data into action. In some cases this entails convincing management that a metric is toxic and should be removed as a gauge of success.
Define and launch sound iterations to the product. The product manager must synthesize all input and ideas into a concrete plan for what to actually do now. This means developing the design, defining requirements, and making tactical decisions pertaining to execution.
Feed the overall learning of their organization. While it's tempting to say that its the product managers job to make their product succeed, some products are doomed to fail regardless of management. If a product manager is responsible for such a product, it is their responsibility to communicate their experience with the product in a manner that makes the company smarter as a whole.