The United Kindom's full and official name is the "United Kingdom of Great Britain and Northern Ireland".
History of the making of the UK
1536 - Act of Union joins England and Wales
1707 - Act of Union unites Scotland and England, together with Wales to form the Kingdom of Great Britain.
1801 - The Irish Parliament voted to join the Union. The then Kingdom of Great Britain becomes the United Kingdom of Great Britain and Ireland.
1922 - Name changed to United Kingdom of Great Britain and Northern Ireland, when most of the Southern counties in Ireland choose independence.
The United Kingdom of Great Britain and Northern Ireland (commonly known as the United Kingdom, the UK, or Britain) is a soverign state located off the north-western coast of continental Europe. It is an island nation, spanning an archipelago including Great Britian, the northeastern part of the island of Ireland, and many smaller islands.
England is a country that is part of the United Kindom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west and the North Sea to the east, with the English Channel to the south separating it from contiental Europe. Most of England comprises the central and southern part of the island of Great Britian in the North Atlantic.
Great Britain comprises only England, Scotland and Wales.

Edit: I know the answer to the question that is the topic subject ... or maybe I just think I do. The question is about flying the flag of the island of Great Britain since Great Britain is not a country. http://www.enchantedlearning.com/europe/britain/flag.shtml
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The issue of the flags available to members has come up before. For instance many members claim to be from the non-country of Great Britain. The issue is described here
http://blog.cgpgrey.com/the-difference-between-the-united-kingdom-great-britain-england-and-a-whole-lot-more/ also here http://www.wimp.com/differencebetween/
... with the amusing introduction "The United Kingdom, England, Great Britain? Are these three the same place? Different places? Do British people secretly laugh at those who use the terms wrongly?"