Private schools: they charge fees and don't follow the national curriculum. They are inspected regularly.
Voluntary Aided Schools: state-funded school in which a trust contributes to the building costs and has influence in the running of the school.
Voluntary Controlled Schools: funded by the government via the local authority. They don't charge fees and are frequently faith schools. (Who decide about the religious content taught).
Academy Schools: state-funded by the Dept. Of Education. Independent from local control. Self-governing non-profit charitable trusts.
Free Schools: independent, non-profit, state-funded (by parents, or charities, or teachers) free from local control and can rewrite the curriculum.
Community School: state-funded. The local authority employs staff and is responsible for admissions. They are not influenced by business or religious groups and follow the national curriculum. It's a place where education, community resources etc. convey to offer opportunities, involvement of families, better learning, health etc.
Grammar Schools: initially born to teach Latin, then focusing on academic studies for pupils who would go on to study at university. They are state secondary schools that select their pupils by an exam called 11plus. The rest go to secondary modern school. More common is the comprehensive system in which pupils of all abilities are taught together promoted by the Labour party since 1950s.
Foundation Schools: funded by central government or by trusts. They don't charge fees.
State-subsidized-schools: partly paid for by the state; subsidized by the state.