The Church of England is also known as the C of E, the Episcopalians (outside the UK), the Anglicans and by other names.
The Church of England was founded because King Henry VIII of England didn't get on with his first wife and the pope wouldn't allow him a divorce.  Believing Anglicans must accept that somehow Henry's marital problems lead their church to divine truth.
The Church of England tend to see themselves as between Roman Catholicism and Protestantism, they don't accept the authority of the Pope and their beliefs are largely Protestant. Most Church of England members in the UK are Liberal Christians
C of E
C of E is a slightly derogatory name for the Church of England used in the UK. Committed members of that church tend to call themselves Anglicans rather than C of E.
English people sometimes put C of E down on forms asking their religion. Some do this because it's traditional or because they identify with the C of E even though they have no religious belief. If you don't want C of E chaplains visiting you in hospital, then talking to you when you are sick and vulnerable never put C of E on a medical form or a hospital admission form.