Britain’s Overseas Territories retain their connection with the United Kingdom because it is the wish of their peoples that they do so. They have a substantial measure of responsibility for the conduct of their own affairs.

Local self-government is generally provided by an executive council and elected legislature. Governors or commissioners are appointed by the Crown on the advice of the British Foreign Secretary and retain responsibility for external affairs, defense and, usually, internal security and public service.

Britain’s Overseas Territories are:

Anguilla
Bermuda
British Virgin Islands
Cayman Islands
Falkland Islands
Gibraltar
Montserrat
Pitcairn Islands (Ducie, Henderson and Oeno)
St. Helena and its dependencies, Ascension and Tristan da Cunha
Turks and Caicos Islands
There are also three territories with no indigenous population:

British Antarctic Territory
British Indian Ocean Territory
South Georgia & the South Sandwich Islands
Additional information can be found on the Foreign and Commonwealth Office website.