While the precise details of how the U.S. manufactures currency are secret, the overall design and manufacturing steps are not. On this page the U.S. Department of Treasury lays out the entire production process, describing the design phase, engraving process, offset printing, platemaking, et cetera. They even describe how the paper comes in on palettes of 10,000 sheets.
More fascinating is listening to Larry Felix, current Director of the Treasury's Bureau of Engraving and Printing, explain how "19th-century craftsmanship" is at the heart of designing and manufacturing currency. Human beings carve insanely detailed images into metal plates (backwards, of course) using hand tools like this:
The craftsmanship required to wield these tools becomes the defacto first line of defense against counterfeiting. "If I were to ask any one of my banknote engravers to engrave the very same design, exactly the same, twice, that person can't do it," Felix explains. "Because of the subtle human nature of designing. The human aspect of engraving a banknote is unique to that individual process and unique to that time."In the following video, you can hear more from Felix and watch a master engraver doing his thing. Be sure to listen for the neat trick he describes, around 2:24, for how to "erase" an engraving mistake—and potentially save yourself from flushing nine months of work down the drain.