Brakes used to clamp onto the seatstays and forkblades so you could adjust them up and down to fit different size rims. What you couldn't do is change the amount of clearance the frame had. So a frame built for 700 outer diameter size tire/wheel combinations could use a 700a, 700b, 700c by moving the brakes.

Originally 650b and 650c had the same outer diameter with 650c using a smaller diameter rim but fatter tire. Now the 650c rim size has been adopted by triathletes using super skinny tires. 650b still sports manly size tires.