An independent school in the United Kingdom is a school relying for all of its funding upon private sources, so almost invariably charging school fees. In England and Wales the term public school is often used for independent secondary schools, and the term private school for independent preparatory schools. Membership of the Headmasters' and Headmistresses' Conference is often considered as what defines a school as a public school, though this includes many independent grammar schools.
Public schools were traditionally a single-sex boarding school, although many now accept day pupils and are coeducational. The majority date back to the 18th or 19th centuries, and several are over 400 years old.
The term "public" (first adopted by Eton College) refers to the fact that the school is open to the paying public, as opposed to a religious school, which was open only to members of a certain church. It also distinguished it from a private education at home (usually only practical for the very wealthy who could afford tutors)
The division of some public schools in to majors or minors is subjective and I don't think there is a definitive list. or criteria.
Nine of the more established schools: two day schools (Merchant Taylors' and St. Paul's) and seven boarding schools (Charterhouse, Eton, Harrow, Rugby, Shrewsbury, Westminster and Winchester) were investigated by a Royal Commission. whose report formed the basis of the Public Schools Act 1868.