The term “Anglican Church,” or in Latin “Ecclesia Anglicana,” originally referred to the Catholic Church in England. In the sixteenth century, for complicated reasons that had to do with royal politics as well as with the ideas of the Protestant Reformation, this “Ecclesia Anglicana” became separated from the See of Rome and adopted theological and liturgical reforms that aligned it with the broadly “Reformed” wing of Protestantism. However, there were English Protestants who wanted to go much further in a Reformed direction, and in the ensuing conflict (which lasted for about a hundred years) the “Church of England” came to define itself over against Reformed Protestantism as a “via media,” a moderately reformed continuation of the historic Catholic Church.
In the 19th century there was a further move in a Catholic direction, so that many Anglicans today will define themselves as Catholic rather than Protestant (while others, such as myself, would say that Catholic and Protestant are not necessarily antithetical, if “Protestant” is understood in a historical sense and “Catholic” is understood as a claim of continuity with the historic Church).
Also, as English-speaking people and English-speaking missionaries spread throughout the world, Anglicanism came to be a worldwide fellowship of churches rather than just the “Church of England.” The American member of the Anglican Communion is the Episcopal Church, although right now there is a huge conflict going on within Anglicanism (primarily over homosexuality, but there are broader issues at stake), which appears to be resulting in a separation between the Episcopal Church and much of worldwide Anglicanism. In the meanwhile, there are a number of Anglican churches in the U.S. which are not part of the Episcopal Church. Some of them are linked to other provinces within the Anglican Communion, while others are not. So when an American calls himself or herself an Anglican, this doesn’t automatically tell you (any more) exactly what ecclesiastical body the person belongs to.
It should be emphasized, though, that this is a fairly recent phenomenon. Compared to other versions of Protestantism, Anglicanism has maintained a high level of unity (many groups have split away, but they have clearly developed into something other than Anglicanism and usually do not claim to be Anglican).
Edwin