I have a 1992 Buick Regal with 53,000 miles. Lately, when coming to a stop, I can feel the brake pedal moving up and down under my foot. It feels really weird.
There are a number of possible causes to a brake vibration/pulsation condition. Typically, when the problem originates in the front, you will feel a noticeable shimmy in the steering wheel while applying the brakes. If the problem is in the rear, the steering wheel will not shimmy, but the vibration will be felt throughout the chassis.
Usually when a slow up and down movement of the brake pedal is noticed, as in your situation, this indicates a severe brake rotor and/or drum problem. Your Regal is equipped with four wheel disc brakes, so forget about the drums. Again, if the brake pedal is bouncing and the steering wheel is smooth during braking then the cause is in the rear. Any steering wheel shimmy in addition to the pedal movement indicates the front brakes.
Get the car to a qualified technician to fully inspect the brakes, front and rear. Chances are you'll be looking at the resurfacing or replacement of one or two pairs of rotors along with the installation of new brake pads.
Hopefully the technician will do a good job of eliminating any possible abnormal causes of the condition. For example: a bad caliper hanging up, rotors which are beyond the minimum thickness tolerance, a loose wheel bearing, rotor corrosion on vehicles that sit for long periods of time, and so on and so forth.
Brakes are simply a maintenance item on any car. Just be sure to take care of problems quickly or they'll end up costing you more money in the long run.