We talk often about Product Managers and believe that their role is very similar to that of a Business Analyst. Like all job titles, Product Manager is used differently by different organizations. Generally they are responsible for eliciting requirements from customers (who may be internal or external), analyzing and documenting the requirements. They also assist with prioritizing requirements and work with the technical solution team to get the requirements implemented properly. Often Product Managers are located within the Marketing and Sales functional areas in a company.
In terms of skills needed to be a Product Manager, they are many of the same skills that we look for in a BA. I would list the ability to see the big picture and also be able to be very detail oriented. They must be good listeners, help customers articulate their needs, and ensure the product is built right and meets customer needs. In many organizations they also manage requirements changes, and are responsible for prioritizing requirements by their importance to many customers. Product Managers often have to make choices about which customers will get their requests first. This is a delicate balancing act!