The definition of a craft brewer is open to interpretation. For some, it is any brewer that is not one of the big industrial brewers like Anheuser-Busch, Miller, or Coors. For others, it is limited to those artisanal brewers who work with natural ingredients, in small batches, and with no automation.
Beer Advocate defines a craft brewer as
One whose primary focus is brewing craft beer, as defined.
It is then important to have a definition of craft beer. Beer Advocate's definition is
Beer brewed in limited quantities often using traditional methods.
There is still plenty of room for interpretation.
"Craft brewer" came into vogue as a replacement for microbrewer. "Microbrewer", then defined as any brewer that brewed less than 15,000 bbl per year, made it easy to distinguish them from the big industrial brewers who brewed millions of barrels a year. However, as brewers like Boston Beer Company (makers of Sam Adams) and Sierra Nevada busted through that artificial limit, a new definition was needed to distinguish "good" beer from mainstream beer. Thus, "craft".