Around ten years ago, if you would ask what the product designer does, you'd hear that it's someone who makes physical things because, essentially, that's what they did.
Product designers nowadays are digital problem solvers (most often). They do research to identify user problems and utilize design skills and available tools to create websites and apps that address them.
Many people think that product designers are UX designers, but there's a slight difference in the roles and responsibilities of the two. Most product designers are UX designers, but very few UX designers are product designers.
UX designers create something from scratch or re-design an existing product. They solve new problems through a user-centered design approach.
On the other hand, product designers improve existing designs and continuously evolve them to create the best user experience. They also make it easy for users to learn about new features and test their designs to create an experience tailored for the users.