1991 Buick Century Drivers door Power Window switch.

You say the driver window works? wouldn't be a main fuse or fusible link then -driver door locks and window wouldn't work either. Each door has a module inside (computer) . The driver door has a circuit breaker power feed an a fuse DDM fuse 10amp located in rear junction block circuit breaker is located in under hood maxi fuse block. The circuit breaker also feeds battery voltage to the other door modules too. PDM - passenger door module fuse 10amp located in the instrument panel fuse box and the RDM - rear door module RDM fuse 10amp located in the rear fuse box - under the back seat.

Each power window is connected to a permanent magnet motor. The motor raises or lowers the glass when the door module applies voltage to the motor. The direction the motor turns depends on the polarity of the supply voltage provided by the associated door module. The 2 front doors have their own separate modules. The rear seat contains a single module, which controls both of the rear power windows.
A serial digital communications link, called the simple bus interface, is used for communications between door modules and the driver's door power window switches. The use of serial digital communications allows the use of a single wire to convey switching information between the power door lock switch and the driver's door module (DDM). The serial communication is also used in order to transmit switching information from the DDM to the other 2 door modules.
The door modules apply battery voltage of the correct polarity to the respective window motor when any local, or power door lock switch, UP switch is operated, driving the window up. Battery voltage is applied to the respective window motor in the opposite polarity when any local, or power window switch, DOWN switch is operated, driving the window down.

Driver Door Module (DDM)
The driver's door module (DDM) directly controls the driver's door window motor and the express down window feature. The DDM sends window UP/DOWN messages received from the power door lock switch to the passenger door module (PDM) and to the rear door module (RDM) over the simple bus interface (SBI) line. The DDM also contains a class 2 data link which communicates with the other vehicle system modules. The left front power door locks and the outside power mirror are also controlled through the DDM.
Passenger Door Module (PDM)
The passenger door module (PDM) controls the passenger's door window motor. The PDM controls the passenger's door window directly from the right front power window switch or from instructions over the simple bus interface (SBI). The right front power door locks and the outside power mirror are also controlled through this module.
Rear Door Module (RDM)
The rear door module (RDM) controls both of the rear door window motors. The RDM controls the rear windows directly from the rear door window switches or from instructions over the simple bus interface (SBI).

Can you control the windows from the drivers seat? Do they operate from the individual seats?
Sounds like the switches may be bad... BUT - CHECK all the fuses...sometimes removing and re-installing the same switches makes the connections fresh again..(I have done this very thing).. and they work

There is no relay for power windows !
Power Window System Components
The power window system consists of the following components:
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Driver window switch

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Front Passenger window switch

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Left Rear window switch

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Right Rear window switch

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Window regulator motors in each of the doors

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PWR WINDOWS PWR SUNROOF Circuit Breaker 30A

When the ignition switch is in the RUN or ACCY position, or Retained Accessory Power (RAP) is enabled and the lockout switch is disengaged, battery positive voltage will be available at the passenger window switches. When an individual window switch is pressed in the down position, battery positive voltage is applied to the window regulator motor causing that window to open. When the individual window switch is pressed in the up position, voltage is applied to the window regulator motor in the opposite direction causing that window to close. The return path to ground is supplied through the inactive control circuit being normally grounded through the driver window switch.

Enter this string in your web browser under the images search bar"wiring diagram for a 1988 buick century window switch"
and you also look at this web page:
http://ricksfreeautorepairadvice.com/window-goes-up-but-not-down-window-goes-down-but-not-up/

i have fixed this problem on mine remove front drivers door panel trace your ground wire from window switch and make sure you have a good ground once you fix your ground you will fix your windows

You most likely have a bad window lift motor or MORE LIKELY a broken window regulator. The door must be disassembled to replace either or both of these.
It could also be a bad window control switch or a broken wire in the door-jamb area. Please note that EITHER of the switches for the passenger window (the driver's control switch or the one on the passenger door) can cause BOTH of the switches to not work to move the passenger window. In this case, electrical diagnosis must be performed to locate the cause of the problem.

probably needs new switches (in driver's door panel). if window is ever left down while raining, switches will quickly corrode.

check to see if the child lock is on.there is a master switch on the door panel

If the master switch that controls these windows you are talking about try the switches on the rear doors first. If they work the windows the drivers master switch needs to be replaced only

Replace the Driver side window switch. These have a corrosion issue.