How do I check passenger window swith - 2003 Oldsmobile Silhouette

Window fuse's and relay's may be up under driver or passenger side dash or behind glove box, under the hood and even under the seats, and finally on the ends of the dash. All passenger power windows and locks are controlled thru and get power from the master control switch on the drivers door.

Drivers door window gets used the most so it could be the motor, window switch, relay or fuse.
Could also be a broken wire in the rubber boot going from the door to the body. You will have to eliminate each till you find the culprit. Check that all switches return to neutral when released.
The motor and the relay are the most likely to cause this problem. Try swapping the relay with a working window relay. If it still doesn't work, remove the door panel and remove the window motor.
Turn the motor shaft in both directions by hand several turns. Then plug the motor in and see if the switch will now run the motor. Sometimes that's all it takes to get it working again. I have had to do this with brand new motors. If voltage is not present at the motor connector, either a bad switch or a broken wire exists. Make sure to check for voltage coming to that windows switch. Actually, GM use's a circuit breaker type relay for each window and one fuse for all the circuit breakers.
Try this....
Open the door, turn the key to the ON position without starting the engine and everything that makes noise is off. Lean down placing your ears close to the bottom center of the door or at the speaker grill. Press the window switch UP and DOWN. Listen for the window motor. If heard the drive is damaged. If not heard, turn key off and locate and replace the relay with one of the other window relays. Check window again. If it works now, get a new relay. More often than not, removing the motor and turning it by hand will get it working good as new. I have had to do that with Brand New motors! So try it.
If all has failed, with the door panel still off, check for voltage at the motor connector when pushing the window switch. No voltage means its the switch, especially if its a auto down or up switch.