Definition of Britain
The 'UK' is short for The United Kingdom of Great Britain and Northern Ireland.
Britain is just a shortening of Great Britain.
Great Britain is made up of England, Scotland, Wales and all the associated small islands. England is the largest populated by far and of course contains London, the capital city of the UK, hence many people outside of Britain speak of England synonymously with the UK. All the inhabitants of the UK are British but the people of Scotland, Wales and Northern Ireland in particular still very much like to emphasise that they are Scottish, Welsh or Irish. Any traveller calling them English is likely to get short shrift!
England, Scotland, Wales and Ireland were of course independent countries in their own right before they were one by one incorporated into the United Kingdom. Ireland (or Eire) is the only one that has reversed this process, becoming the independent Republic of Ireland in 1921, with only the north west, Northern Ireland, remaining under the jurisdiction of the UK government. Scotland, Wales, and Northen Ireland have more recently had some autonomy in their government handed back but they still remain within the 'Union'.
England, Scotland, Wales and Northern Ireland all retain their own capital cities. London is of course the capital city of England as well as the UK, Edinburgh is the capital of Scotland, Cardiff the capital of Wales and Belfast of Northern Ireland. Dublin is the capital of the Republic of Ireland.
There is another term, the British Isles, which is a geographical definition, rather than political. This includes Great Britain, Ireland, the Channel Islands (Jersey, Guernsey, Sark, Aldernay and Herm) and the Isle of Man. The Channel Islands, which are off the northern coast of France, and the Isle of Man, off the west coast of England, are self governing British Crown dependencies.