When the term 'power brakes' is used, what is generally meant is power-assisted brakes, in the manner as power steering is actually power-assisted steering -- it still requires force on the part of the operator, but that force is multiplied by components of the engine.
I recently did a major brake job on my econobox of doom, so I was schooled on many of the major parts of a power brake system and how they work. They are usually:
- Pedal - No great explanation here. It's the thing you depress with your food to arrest forward movement -- the 'interactive' part of the braking system.
- Booster - This is what gives the 'power' in 'power brakes'. Generally the booster is a large air_tight bell with one or two large diaphragms. A rod runs from the brake pedal on one side to the master cylinder on the other, straight through the diaphragm, where it it is attached.
- Master Cylinder - responsible for the movement of the brake fluid through the brake lines. WonkoDSane has a great writeup in that node detailing how they work.
- Brake Line - either metal or stress-wound rubber lines that run from the master cylinder to the individual wheel cylinders.
- Wheel Cylinder / Slave Cylinder - the opposite of the master cylinder in that it converts the fluid pressure into physical movement.
- Calipers / actuators - they come in many styles, but are always the thing that pushes the brake pads / brake shoes against a brake rotor / brake drum to slow the movement of the vehicle.
How it all works together
Okay, so I'm about to hit someone's puppy that is wandering aimlessly in the middle of the road, and so I jam my foot on the first element of the system, the brake pedal. This does two things -- it moves the rod attached to the diaphragm inside the booster, and it also activates a few vacuum valves. In this case, the valve on the side of the diaphragm closest to the master cylinder opens, and the air inside that side of the diaphragm is evacuated and it moves toward the master cylinder. Since there is a rod attached to the diaphragm which in turn it attached to the piston in the master cylinder. This give many times more force than the human foot could normally produce.
From here, the master cylinder pushes brake fluid, under great pressure, toward the individual slave cylinders, which in turn move the acctuation system on that kind of vehicle, be it drum or disc brakes, which cause the brake shoes or pads against the rotor or drum which creates drag and slows the car down.
When you let off the brake, the exact opposite happens -- the valve on the other side of the booster opens and the air in that side is evacuated and in turn makes the piston on the master cylinder retract and therefor release the presure on the slave cylinders and braking stops. This allows better braking response times.
However, I didn't hit the brakes in time and I killed your puppy. Sorry. | 
