Why does Mazda Protege shimmy when braking?

Odds are it's probably not just your Mazda. Most vehicles do that when the rotor is shot or warp\ped. The rotor is the big disc (or cd) that the disc brakes apply pressure to when you push your breaks. When the surface is warped or shot, the surface is no longer smooth, causing what can be a shimmy shake. You can tell by taking off your tire, and by rubbing your finger up and down the rotor. If you feel a lip or rough edge from the edge towards the inside, then it's the rotor. They are generally about $20-$30 to replace. Otherwise it could be that one or both of the wheel bearings are shot.