In common British usage, a school open to the public that charges fees and is financed by bodies other than the state, commonly as a private charitable trust; here the word "public" is used much as in "public telephone". See Public school (UK).
In the United States, a school which does not charge tuition fees but is financed and controlled by the government, in contrast to a private school[?] (also known as an independent school); here the word "public" is used used much as in "public library", i.e. to mean "provided to the public at public expense". There is a small but increasing movement to leave public schools in favor of home schooling, which is learning at home.