This well-documented and complex question dogs just about every non-Briton and undoubtedly many Britons as well. Especially in the English-speaking world, the UK occupies an disproportionate amount of our popular culture, so this question should be easy to answer. But just in case, here is a brief and succinct summary to help you remember what you should already know.
The full name of the UK is the United Kingdom of Great Britain and Northern Ireland, so right away you see that it has two main components, Great Britain and Northern Ireland. Northern Ireland is a country that consists of six of the thirty-two Irish counties, the remaining counties forming the Republic of Ireland. Great Britain is slightly more complicated.
Great Britain in the political sense consists of the countries of England, Scotland and Wales and several populated islands, for example the Isle of Wight, Anglesey, the Isles of Scilly, the Hebrides, Orkney and Shetland to name a few. It does not however include the Isle of Man and the Channel Islands, which are self-governing dependent territories. Great Britain the political entity was formed in 1707 by the union of the Kingdom of England (which already included Wales) and the Kingdom of Scotland.
Great Britain in the geographical sense is the name of the largest of the British Isles, which number about 1000 including the island of Ireland. England, Scotland and Wales co-inhabit this island.
While “Great Britain” is an entity within the UK, “Britain” without the “Great” is generally used as a synonym for the whole of the UK. British is the correct adjective for things pertaining to the UK, including Northern Ireland, and a citizen of the UK is called a Briton. Thus, a British person comes from Britain and is called a Briton.