"You know, [if] you're a writer, there's your script. You're a director.. but if you're the producer, what did you do? You see, nobody knows what you do. The producer, I mean, all he's got is the credit. You see? And some plaques on the wall. They don't know what we do. Don't get me started." -- Stanley Motts, Wag the Dog
This might seem like a glib answer, but the producer is really the one who is responsible for producing the game. As in, doing whatever it takes to make sure the game actually gets made on time and on budget. For this reason, the producer is typically the highest authority on the game team. Usually the producer's job involves a lot of management of workers, scheduling, setting milestones (which is part of scheduling), staffing decisions, coordination between the different teams working on the game, negotiations with publishers (if applicable), external companies (subcontractors, translators, etc) and generally making sure that everyone on the team is able to get productive work done and is communicating with each other.
To be effective, a producer must generally be one of the hardest-working people on the team, but to a casual observer or team-member they will usually appear to be doing almost nothing, because their work does not directly touch on the game itself, but instead facilitates other people's work... and if they're doing their job well, then things tend to go relatively smoothly. If not, then things tend to go bad really quickly.
It's a very difficult and very thankless job. And at the end of a project, if you've done your job right, nobody typically knows what you did.