Like most Americans, I didn't understand this until I looked it up shortly before going to Scotland, and the "Brexit" discussion does not help because as usual, actual usage is inconsistent.
- Great Britain is the big island containing England, Wales, and Scotland.
- England is a nation made up of the part of Great Britain that is not Scotland or Wales.
- The British Isles includes Great Britain, technically Ireland and thousands of other smaller islands.
- The United Kingdom is made up of the nations of Great Britain, plus Northern Ireland.
Strictly speaking, saying "Britain" was voting on the EU exit yesterday doesn't really make sense. It was a UK vote. Most of the time when people talk about "Britain" in terms of politics, they mean the UK. In terms of culture, "British" generally means Great Britain (not Irish).