The main types of school:
Community schools - Fully controlled by the LA. These are not faith schools.
Voluntary Controlled schools - some faith schools fall into this category. They are controlled by the LA but a foundation or trust has some influence. For some of these the LA includes faith-based criteria for admissions. The foundation or trust appoints a minority of the governors. Most of these are CofE schools.
Voluntary Aided schools - faith schools with more independence because the foundation or trust owns the buildings and contributes to building costs. These schools set their own admission criteria and generally include faith-based criteria. The foundation or trust usually appoints the majority of the governors. Most of these are CofE or RC schools but there are a few of other faiths.
Foundation schools - A kind of halfway house between VA and VC schools. As with VA schools they set their own admission criteria. As with VC schools the foundation or trust usually only appoints a minority of the governors. Most of these are non-faith schools.
Academies - directly funded by the government and independent of direct control by the LA. Most academies are existing schools that have converted to academy status. Some are supported by a sponsor but most new academies are not. An academy cannot be selective unless it has converted from a grammar school. It can, however, have a specialism and select a proportion of its pupils by aptitude for the specialism. They have more freedom than other schools over the curriculum although they must follow the National Curriculum for maths, English and science, and there are also some other restrictions. They are free to set the pay of their teachers. They also set their own admission criteria, although they must comply with the Admissions Code which limits what they can do. Most academies are non-faith schools but a faith school can convert to academy status.
Free schools - basically an academy set up from scratch or converting from the independent sector as opposed to converting from a pre-existing state school. Most free schools are non-faith schools but a variety of faiths have started free schools. A free school cannot be selective on academic ability unless it has converted from a selective independent school.
City Technology College - there are only handful of these left, most having converted to academies. They are very similar to academies but they must have a sponsor who is expected to provide more funding than is the case for an academy. Unlike academies they are not bound by the Admissions Code. They specialise in technology and practical skills.
There are some areas of the country where state grammar schools continue to exist. They select on academic ability. They also fall into one of the above categories.