London is the capital of England and the United Kingdom – but not Great Britain. Granted, this can appear confusing.
England is a country in its own right. London is its capital.

Great Britain is more of a geographical term, referring to the largest of the British Isles (the ‘Great’ in reference to it being larger, rather than: what a 'great' country that is). Great Britain is comprised of England, Wales and Scotland.

The United Kingdom (of Great Britain and Northern Ireland) – consists of England, Wales, Scotland and Northern Ireland. Being British – refers to all UK member countries, not just Great Britain.

The British Isles is again, a largely geographical term as it includes the United Kingdom, the Republic of Ireland and many smaller islands.
The United Kingdom is also a member of the European Union, though retains its own currency (the Pound Sterling).

Why so complicated? History is the reason. With ever changing boundaries, borders and autonomies over the last 2,000 years, the naming has evolved in accordance with the history.