The independent sector is made up of fee-paying schools. The 'elite' independent schools (or, more accurately, their head teachers) are members of an organisation known as The Headmasters' Conference. Such schools are referred to as 'public schools'. (i.e. all public schools are 'independent' but not all independent schools are 'public'). Most public schools (unlike the majority of schools in the independent sector) are primarily (or wholly) boarding schools.
The name 'public school' is clearly inaccurate but, like so many things in this country, it's rooted in history. Eton College, for example, (one of the UK's so-called 'top' schools) was founded by Henry VI, in 1440, as a charity school for poor students.
The average annual cost of education in an independent school is currently �9000 ($18000) for day pupils or �20000 ($40000) for boarders. The top public schools, such as Eton, Charterhouse and Harrow, charge nearly �25000 ($50000) per year.
Chris