what is the american equivilent of private schools?

A public school is run by the state and is open to all students who live within its boarders. Every child in the US has a public school they can attend for free up till senior in high school.

A private school, tuition is paid for by the parent. Most, but not all, private schools in the US are religious schools. Most do not have dorms, students still live at home. I sent my son to a Christian school that was affiliated with a Baptist Church.

There's very little difference - in America children can go to private or state-funded schools just as in the UK.

It's worth nothing that in the UK what we think of as private (independent and non-directly-state-funded) schools are sometimes called 'public' schools, owing to the fact that these schools are descended from the first grammar schools, which were originally maintained at the public's expense, but have developed into fee-paying schools. In the US a 'public school' would always mean a state-funded one. This often causes confusion.

For more you could look here: http://en.wikipedia.org/wiki/Private_school#United...