Actually, asI was taught, the Bobber and the cafe are essentially the same thing. Just different sides of the pond.
After WWII, soldiers returning home to America wanted to ride the same bikes they rode in the war, only to find that the Civilian models had to much stuff on them. So, they started "Bobbing" everything off of them that was not needed in order to go faster. So is born the Bobber.
Soldiers returning home to Europe wanted the same thrill that they got flying. So, they took stock bikes, and like their American Couterparts, started cutting everything off that was not needed in order to go faster. The Euopeans took this a little further by adding lowered bars for less wind resistance, putting high powered engines in better frames (ie the Triton), and for some a fairing to help cut through the wind. They would race them at Cafes or Cafs nad thus the cafe racer.
Choppers were a late 60's early 70's idea for not sticking to the norm and doing their own thing. They would use longer suspension, "Chop" the frames and add more rake, and use wild paint jobs. Thus the Chopper.