Production students focus on the making of film/TV. The camera, the lights, writing, directing and the like.
Producers can and do some or all of the following: writing, hiring crew/writers/actors, budgeting, marketing, getting financing, finding material, finding a distributer, filling out festival paperwork, dealing with studios, dealing with networks, often times dealing with both at the same time, making sure the who crew has everything they need to make a film, making sure everything is running smoothly, making sure we get the shots we need...it's really a long list of things. Basically if it's going to happen, producers make it happen.
So what is a producer? We make movies/tv/web series happen. This is why producers are awarded the Best Picture Oscar.
We also often don't share the spotlight, we make it happen and then we let others take the stage. For example do you know who Emma Thomas is or Kathleen Kennedy? Thomas produces the Batman films and Kennedy work as a producer for Steven Spielberg and now has her own company and more recently was named co-chair of George Lucas's company.
I think it's interesting how the public doesn't normally know these people, but they are highly influential when it comes to making TV and film. Not that this is a bad thing, until I got interested I didn't know many producers. I feel we're sort of like book editors, we make things happen but unless you're in the field you often don't even know to thank them.
Out of curiosity what is your favorite film and who was the producer? (Check out IMDB.com for help looking up crew)?