In the UK, a private school is a school for which the pupils must pay to attend. They are usually home to snobby, stuck up kids with a supiriority complex, even though tuition in such establishments is not much better than in the free alternative.
Very prestigous private schools are known as public school
The Art's Educational Secondary School in london is a private school, and Eton is a public school.
A pupil at a private school is known as a private schoolboy/girl, and a at a public school is known as a public schoolboy/girl.
A free school in the UK is known as a state school