They're all over the country. I typed in "Governor's School" and found respectable sounding math/technical schools. But I also saw personal horror stories from kids on their home pages. One was of the NC Governor's School.
Apparently they're run by the state board of ed. Be very wary. It wouldn't hurt to talk to alumni. Maybe you could ask for an alumni list?
It looks to me like they are choosing the best and brightest for indoctrination, the future leaders of our country.
Everyone reading this should be VERY frightened for the future of our country. I wasn't even aware of the exsistance of this school and I venture to say most people aren't.