Private schools are independent, privately funded non-state schools that are not financed by federal, state or local government. Children who attend private schools may be there because of wealth or financial sacrifices of their parents, and may be selected for their academic prowess, or prowess in other fields like athletics or the arts or sometimes their religious background such as Judaism or Roman Catholicism.

Private schools retain the right to select their students and are funded in whole or in part by charging their students for tuition, rather than relying on mandatory taxation through public (government) funding; at some private schools students may be able to get a scholarship, lowering this tuition fee, dependent on a student's talents or abilities (ie sport scholarship, art scholarship, academic scholarship), need for financial aid or tax credit scholarships that might be available. 

Private schools are generally exempt from most educational regulations at the Federal level but are highly regulated at the state level. These typically require them to follow the spirit of regulations concerning the content of courses in an attempt to provide a level of education equal to or better than that available in public schools.

public school vs private school
Some public schools, like charter schools, offer the perks of private schools (flexible curriculum, regulation exemption) without the cost of tuition. The biggest perk of attending a public school over a private school is saving money. Public schools are free—no tuition required.

The other big difference between the public and private school class size, and student-to-teacher ratios, which is considerable. In public schools, the average class size is 25 students, compared to 19 students per class in private schools according to the National Center for Education Statistics (NCES). 

Another perk is that private schools tout is a diversity of classes. Since they are not under state supervision, they can offer a curriculum that suits their focus. If you have a child that wants to study the arts, theater, music, or other such subjects a private school will probably be a better fit