All of a sudden one of the toilets is making a horrific noise after we flush it. It sounds like the feedback you would get from a microphone when you stand in front of a speaker or amplifier. The other toilet is in the next bedroom and does not make any noise. It doesn't sound like it is coming from the toilet tank but from the pipes. It is really loud and happens after the toilet is done refilling.

Hi spearmint and welcome, if you look to the right side of the screen you'll see Similar Threads, your solution may be there, if not post back and someone will give you more help.

That sound would be caused by either a defective washer either in the fill valve or the stop valve under the tank. Assuming a Fluidmaster fill valve.
How to fix a toilet: replacing the Fluidmaster 400 fill valve seal (HD) - YouTube