In general, most US high schools are separated into the following categories:
Public Day School, Private Day School and Private Boarding School. There are other specialized schools, like military, special education, religious, etc.
When your child chooses a school, ask the following:
1) Big or small
2) Countryside, city or in between
3) All boys, all girls or co-educational
4) What sports does the school have
5) What extra-curricular activities does the school have
6) How strong is the school academically compared to the level of your child