The United Kingdom of Great Britain and Northern Ireland is the official name of England, Wales, Scotland and Northern Ireland, but this is too long for practical purposes, so it is usually known by a shorter name. At the Eurovision Song Contest, the United Nations and in the European Parliament, it is referred to as 'The United Kingdom'. In everyday speech, this is often shortened to 'The UK' and in internet and email addresses it's '.uk'.
In other contexts, it is called 'Great Britain'. For example, during the Olympic Games and in international banks where the term GBP (Great Britain Pounds) is used.
The normal everyday adjective, when talking about anything to do with the UK, is 'British'.