(Note: This post is meant for geographically challenged American readers.)
The British Isles
The British Isles are the group of islands northwest of continental Europe. The main islands are Great Britain and Ireland, and there are thousands of smaller ones.
Great Britain is the largest of the British Isles, comprising the countries of England, Scotland, and Wales. It was originally named Great Britain to differentiate it from Lesser Britain, which denoted the region of modern-day Brittany in France. The name Great Britain stuck even after Brittany was incorporated into France.
Great Britain is sometimes shortened in speech and writing to just Britain, and Britain is also sometimes used as shorthand for the United Kingdom as a whole.
England is the largest country on the island of Great Britain. It borders the country of Wales to the west and the country of Scotland to the north.
The United Kingdom
The United Kingdom is a constitutional monarchy comprising England, Scotland, Wales, and Northern Ireland. This is where things can get confusing. Scotland, Wales, and Northern Ireland have governments of their own, although they all fall under the power of the parliamentary system with its capital in London. The state also holds three Crown Dependencies and a number of overseas territories that are not constitutionally part of the U.K.
Unless you wish to cause offense (or offence, in British English), referring to Welsh or Scottish people as British is not a good idea. Foreigners can usually be forgiven for the lapse in decorum, but Welsh and Scottish are preferred. There are some complex historical issues behind these names, and we should always try to call people what they call themselves.
Of course, the Republic of Ireland (which shares the island of Ireland with the U.K. country of Northern Ireland) is not part of the U.K., so to refer to people from Ireland as British or English is wholly inappropriate (even though Ireland is in the British Isles).