As I had to Google it - apparently BARF is "Biologically Appropriate Raw Food".
I have to say I get the concept. Dogs just didn't evolve to eat the weird hard little grain-based pellets that a lot of people feed to them. Then again, humans didn't evolve to eat a modern Western diet either. Considering the number of fat and unhealthy dogs I see is starting to rival the number of fat and unhealthy people, perhaps that tells us something.