The brake booster is what gives the brake their power and helps reduce the force the driver must exert against the brake pedal. In most vehicles, it uses vacuum from the engine to push on the master cylinder, via the pressure applied to the booster when the driver steps on the brake pedal.
The brake booster usually fails due to wear over time, or exposure to oil from a leaking master cylinder. When the brake booster fails then it should be replaced.
Most manufacturers will cover the brake booster under the bumper-to-bumper portion of the factory warranty if it fails due to a manufacturer defect or workmanship. There are extended auto warranties available that will cover the brake booster under certain levels of coverage.