I take it you mean the big black round bit? That's the brake servo, also known as a "brake force amplifier". I believe its principal of operation is to take a vacuum from the engine and use it to assist the brake hydraulics somehow, with the assistance being proportional to the pressure on the brake pedal.
Now you mention it, I think I'm going to have to look it up and find out exactly how it works
Anyhow, the whole engine vacuum thing is the reason you only get about 2 assisted presses of the brakes with the engine off, before braking involves pushing HARD into your chair (yes, I've made that mistake in a traffic jam downhill...

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<steve_earwig> I think this forum is more about keeping our cars going with minimal outlay than giving our cars more reason to go bang