Producer is a catch-all label for various roles in games to do with management, planning, commissioning, evangelising and funding game projects. Its meaning varies somewhat from company to company, but generally speaking there are two types of producer: internal and external.
Internal producers work in game studios as managers of teams. They are usually the boss in terms of project management, costs and human resources issues, but also commonly have a creative or quality assurance role. In some studios there is a separation between the producer and a game director, where the producer has no creative involvement. In others, the producer is also the creative lead. Internal producers are often considered most accountable for the final release of a game and they report to either executive producers or studio heads.
External producers work in publishers rather than developers, and are involved in comissioning and stewarding games toward release. An external producer usually would not play a part in the day-to-day business of how the project runs, but is most responsible for approving milestones, organising quality assurance on the publisher side and managing payment schedules to developers. External producers are also commonly involved in financial forecasting, budgeting and planning across a number of projects rather than just one.
Some other kinds of producer are: associate producers (essentially assistant producers), independent producers (individuals who assemble teams and funding per-project), project managers (who only deal with work flow), executive producers (the bosses of other producers) and studio heads.