This is a common question as the role can seem ambiguous and there are certainly no shortages of answers on the web. However, all of them focus on the results and are top-down oriented. "Product managers define the product." "Product managers keep everybody on the same page." "Product managers tell support how to troubleshoot."
These are not untrue but there are subtle differences that can make or break a team and it all comes down to the title, Product Manager. Strictly speaking, this means the person manages the product; it doesn't say anything about managing people. So, if you've got a great team to build your product (this is an assumption, I realize) and you want to get your product done, what's your best course of action? Get out of the way!
There is still plenty of work for you to do. Complications will arise. Human nature says somebody will inevitably have a bad day. Human nurture will translate that as a personal affront. Chaos theory tells us it could all lead to an uprising of a whole department, dogs and cats living together... mass hysteria! Unless we step in.
But not to take over. We step in to remove impediments. We step in to help the team. We step in to keep the team highly functional. We step in to serve the team.
This can take on many forms. Engineering doesn't know what to work on and the customers can't describe it so product management will define the scope. Sales already sold it; marketing is creating the release kit for next month; QA wants three more months to thoroughly test. Product management eases the burden and miscommunication of all departments by keeping everybody on the same page. Support spends time with a customer learning how to troubleshoot a new product. To keep support high functioning, the product manager will preemptively educate the support group.
Do the end results sound familiar? It's the difference between management and leadership; power and authority; fear and respect. There are certainly highly-successful teams on both sides of the spectrum but which would you rather work for?
What do product managers do? They do whatever it takes to help you and everybody else in your organization focus and succeed in making exceptional products.