Buick 1990 Century the bottom pulley broke and the outer ring came off. how do i keep the pulley from spinning to take the center bolt out? And is it the harmonic balancer?

Hello topher
There is a special tool that is used to hold the flywheel on the transmission end of the engine under a removable cover..
The GM tool # is J-97036.
Many aftermarket mechanics would improvise something to hold it as they may not have the tool or easy access to it.
You will need to replace the harmonic balancer as an assembly.
It was a common problem on these car models at that time.
It would also most often damage the crankshaft position sensor when this happened.
Have a close look to make sure the sensor is not damaged from the loose balancer.
The sensor will also need to be adjusted to allow clearances on the tone ring on the balancer, there is very little allowable space. Careful, do not bend the tone rings on installation of the balancer.
Rotate VERY slowly and observe that the tone rings do not contact the crank sensor, adjust sensor as necessary.
The bolt on the balancer has a torque specification of 104 lb ft plus an additional turn of 56 degrees.
I hope this helps you along towards your repair.

KL

Yes this is known as the harmonic Damper and your problen is your missing a part thats why its spining a piece of metal that goes in that area that keeps it from spinning on the crank shaft you need to talk to a mechanic for hands on work