[edit] I read my previous entry and didn't agree with it
The Church of Ireland is Anglican. It was set up when the English were over there (I think), and has the same doctrines etc. as the CofE. It just happens to be in a different country.
The Anglican Communion is a world-wide organisation of Anglican Churches. There is no single "Anglican Church"; it is better to speak of the Anglican Communion, which consists of national churches in communion with the Church of England. Most share the dogma and doctrine of the Church of England.
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The term Anglican describes those people and churches following the religious traditions of the Church of England, especially following the Reformation.
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Anglicanism is most commonly identified with the established Church of England, but Anglican churches exist in most parts of the world. In some countries (e.g., the United States, Scotland) the Anglican church is known as Episcopal, from the Latin episcopus, "bishop", which comes from a Greek word literally meaning an "overseer."