Fundamentally, the PM's the one who gets the product shipped, whatever that entails. If the product is shipped, the PM makes it a success, including shipping followup features or products.
I've heard a few general descriptions, from "does anything except write code" (not true in many circumstances) to "mini-CEO" (implies greater degree of authority than often present). The best one that's stuck is "captain of a pirate ship"; you have to figure out where the ship is going, influencing without authority.
On a general level, rough job scope can be in the "design and plan" categories through to execution, driving everything forward, getting approvals and making sure all the edge cases are covered (both in the product and overall, e.g. legal approvals, trademark searches, etc) - plus measuring success, and making sure those metrics are defined and measurable in the first place. Listening to customers, or organizing user studies before launch, and incorporating feedback at every turn.
You sit in the middle of the product vision, and while engineers might want to make decisions that seem like good ideas from a technical standpoint, you are the guardian of the context which makes them into great ideas (or not).
Similarly, feeding back technical issues and restrictions into the UX design process, and providing those intuitive leaps when you realise if the design goes like this, and the technical architecture like that, the whole thing can be done in a tenth of the time to 99% of the effectiveness.
My impression is that it is a very hard role to describe because it includes "whatever needs doing" and will vary by product and the team around you. It will include making sure everything gets done and tied together including not just the product itself but the documentation and any licensing agreements and ensuring correct marketing information is prepared for the product at the right time (and the marketing team understand the product). It may include what is normally considered business development and associated liason with legal. It may include booking and preparing for a trade show.
In a big company playing a linking role and getting the right people talking can be important even if you then step back.
Very importantly it should be someone able and prepared to take decisions that might not be big and require senior management but that need making and aren't really either engineering decisions or design decisions.
I consider myself a PM and this is what I do:
Srum Lead
Sprint Planning / Release Management
"Tier 3" Support
Documentation Editor
Question Answerer Guy (Why does this piece of software do this and why is this link not blue?)
I'm also the liaison between management and the engineering staff.
There is some more official type stuff on wikipedia. http://en.wikipedia.org/wiki/Product_manager
They may not do all the work, but they're the ones covering all the little details: making sure an email goes out when a new user signs up, reviewing that copy, watching retention, etc.
You'll see a lot of job descriptions for product managers reference 'Mini-CEO', which is pretty apt.