An oddity of the British educational system is that independent schools that charge fees are called Public Schools, confusingly they are also often referred to as Private Schools. The sons and daughters of the upper classes and the wealthy establishment typically inhabit such schools.
This confusing situation arose because of how the education system was first established in Britain. The middle ages saw the foundation of Grammar Schools by the Church or through acts of charity. These schools were widely reffered to as public schools and were free. But the students of these schools largely came from wealthy families rather than the general populace. Competition saw the best schools attracting many students and hence they decided to start charging fees.
It was only later that the Government passed an act to allow schooling for all children to be funded by the state, so the general public attend what are now termed State Schools.