Yes it IS a bit complicated! The UK and its constituent parts probably have the longest continuous history in the world without a revolution of some kind apart from maybe Japan - the English Civil War of 1642-48 is probably the nearest we ever came to it - so nothing about this part of the world is ever quite that simple.
Great Britain is the name of the island that includes England, Scotland and Wales. The United Kingdom used to include the whole of Ireland as well, but it got independence in 1922 except for Northern Ireland. The reason for the split is that those six counties contain a majority that wants to stay part of the UK, while the rest didn't.
If I may digress a little, that is a long and bloody story too long to put here and has resulted in a lot of recent terrorism in the campaign by nationalists for Northern Ireland to join the rest of Ireland. When I was younger, going into central London always carried a slight risk of being blown up by Irishmen. In 1983, they almost did - I was shopping in Harrods when they exploded a car bomb outside. So I have a slightly different view on American fears of terrorism.
The full name of the country is the United Kingdom of Great Britain and Northern Ireland, and that's what it says on the cover of my passport. So having defined Great Britain, there's your answer to question 1.
The problem with the name of the country is it doesn't lend itself easily to forming an adjective. The USA has the same problem - I mean, "United Statesian"? It doesn't work. We have the same problem. So people from the USA are American, and people from the UK are British. OK, that ignores the Northern Irish but it's the best we could do. Back to my passport again - my national status as printed in it is "British Citizen".
That answers the questions but here are some more random facts which will hopefully fill it out a bit.
Great Britain - isn't there a Little Britain? Yes there was - it is now the French province of Brittany. In French, Brittany is "Bretagne" and Great Britain is "Grande Bretagne". Brittany actually has a historic Celtic language all of its own - Breton - which has no relation to French whatsoever, but is very similar to Welsh (which in its turn has absolutely no resemblance to English, which is a branch of West Germanic and is most closely related to Dutch and German, according to linguistics experts - the main differences are because of the Norman Conquest of 1066, which caused English to absorb a lot of French words). A couple of examples - good morning in Welsh is "bore da", pronounced "borreh da". One that most British people know if only from having seen it on signs is "Welcome to Wales", which in Welsh is "Croeso i Gymru", pronounced "croy-so ee gum-ree". See what I mean?
Timeline:
About a thousand years ago - England finally became one country forged out of a lot of little kingdoms.
About 700 years ago - England took over Wales.
1603 - Queen Elizabeth I died and her nearest relative just happened to be the King of Scotland, so he became King of England as well. But England and Scotland remained independent: they just happened to have the same king.
1707 - Act of Union between England and Scotland, creating the United Kingdom of Great Britain.
1800 - Act of Union between the UK and Ireland, creating the United Kingdom of Great Britain and Ireland.
1920 - Act passed to make Ireland an independent republic except for Northern Ireland, to take effect in 1922, thus creating the UK as it is now.