Great Britain is the larger island and contains England, Scotland and Wales.
The UK is actually the United Kingdom of Great Britain and Northern Ireland, but that's a bit of a mouthful so it's usually shortened to UK or Britain for convenience.
Although England, Scotland and Wales are countries in a historical sense, none of them are sovereign countries like say Germany, France or the USA. They are all parts of the sovereign country called the United Kingdom.
Despite what 'dont*shout' says, England is not considered a country on its own. All are parts of the UK and the same status.