RE: 2001 Buick Century power booster

disconnect the vacuum hose to the booster, then disconnect unbolt the master cylinder from the power booster(2 nuts) remove the cylinder from the booster by sliding it off the studs. inside the car under the dash disconnect the brake pedal from the booster rod and undo the booster mounting nuts/bolts there is 4 or them.

Seems like a vacuum leak. Either the valve, hose to the manifold or the booster itself is leaking air.

These are not rebuild-able, they must be replaced if they leak.
One that leaks will hiss while the engine is running. The hiss is vacuum leaking, not pressure leaking out.
To possibly check it under the hood, be very suspicious of a leaking vacuum line at or near the booster, not the booster itself. It might be bad, but that is pretty rare.
The last one I changed I thought was defective because it hissed when I applied the brakes, but a new one did the same thing! :-(
If you decide to replace it, you will be unbolting the master cylinder from the unit under the hood, and also unbolting the unit under the dashboard.
Of you are really careful, you might not need to disconnect the master cylinder 's brake lines at all, just gently pull/push it out of the way to change the booster.
Be really careful not to kink the brake lines.
The unit is bolted to the car with usually 2 bolts through the firewall behind the brake pedal.

NO. the vacuum is used to make the brake pedal easy to push, power brakes. the pedal would be hard without vacuum, that is why the pedal gets hard with the key off and engine not running. the problem you are having is more of a hydraulic problem.
Have the brakes checked asap. the brake system may have a leak causing the pedal to go low. check the fluid level and if it is low or empty then there is a problem, could be a rotted brake line leaking or a wheel cylinder leaking.

nope you have to replace them.when brake booster diaphram leaks you cant plug them.i have bought one at salvage yard in good shape.until i got a new one.get your brake booster from inside salvage yard where they stock them inside out the weather.dont take any brake booster off old outside cars been in all kinds bad weather. there vacuum diaphram probably dried up will leak also.

If it takes time to stop the car and the pedal is to the floor when braking, YES replace the booster

It sounds like you have a bad mastercylinder cause when you panic brake it doesn't have time to bleed through the valve in the cylinder but under normal braking your foot is on pedal longer in turn giving the fluid time to bypass the valve in the mastercylinder in turn the pedal going to the floor.

It probably is from the brake booster. The booster uses engine vacuum to assist in braking. It's either a vacuum hose leaking, or possibly the diaphragm in the booster itself. Apparently, the leak is small right now and is not affecting performance (engine vacuum is sufficient to make up for the leak), but eventually the leak may become large enough to lose the assist. Check the large hose that runs from the intake manifold to the booster for any weathering/cracks, and replace it if there is any doubt. If that is not the problem, you will probably have to replace the booster.

Sounds like your Power Booster is bad.

you have two problems here, your master cylinder has a leaking seal, the low pedal is resault, the power brake booster rubber diaphram is damaged and leaking from the brake fluid that is getting into it from the master cylinder, these are sold as a unit that has been rebuilt.