Academies are state-funded schools that are funded directly by the government rather than by the local education authority.
They were originally introduced under Tony Blair's Labour government (although were similar to grant maintained schools introduced by Margaret Thatcher) but were expanded very significantly by the Conservative Michael Gove when he was education secretary.
The schools can set their own admissions policies and do not have to follow the National Curriculum. Most English secondary schools are now academies, and there are a significant number of primary academies too. They are run by Academy Trusts and some are parts of large chains.