Great Britain is the official name given to the two kingdoms of England and Scotland, and the principality of Wales. The United Kingdom (UK) includes these areas as well as Northern Ireland. Britain refers to only England and Wales, though some will shorten Great Britain with this term.
Correction, flag of the United Kingdom. The flag of Great Britain doesn't have the St Patrick cross as Ireland was not fully incorporated under the crown until 1807 I.e the formation of the United Kingdom of Great Britain and Ireland. Wales is not featured because it's a principality, not a kingdom.