Well, there are differences in every country. If we discuss K-12 education (not college or university) in the United States, independent school districts manage this function supported mostly by real estate taxes at the local level. These are public, nominally free. Private schools are supported by tuition from students, so these schools are expensive. Government is not involved in the funding and management of these private schools in the US.

Private schools are run by governments and everything is bought or used from the money given by the government, whereas individual schools are totally not related to governments and everything is provided from the money from student admission fees and/or monthly fees. The environment in private schools are usually manner-less and the monthly and yearly fees are relatively very lower.

In the UK, this is quite a complicated subject, especially since the current government is making reforms to to school sector. In the UK, there are state schools (run and funded completely by the government and controlled through the local authority), and private schools (wholly independent of the government, funded through donations and tuition fees paid by parents and sponsors) - confusingly, the most prestigious private schools (Eton, Rugby, etc) are referred to as Public Schools, due to a historic naming convention. To further complicate things, there are also new types of school introduced by recent reforms - Academies (independent of local authority control, funded by central government) and Free Schools (run by parents, businesses, charities, and eligible for state funding as well). In addition there are some historical types of school that are less common but remain - for example Grammar schools, which normally have some government funding and control but which have strict selction polices. But when people refer to the "independent sector", on the whole they are talking about private schools.