There is a sovereign state called the United Kingdom of Great Britain and Northern Ireland. You would understand this as "Britain".
It has a number of much smaller territories under its protection, relying on the UK for defence and international representation, but they have almost complete autonomy and act like little countries. Those are the "overseas territories" and include places like the Falkland Islands and the Isle of Man, as well as barely-inhabited Antarctic places like the South Sandwich Islands.
Great Britain refers to the largest island in the British Isles before Ireland. In the island of Great Britain there are three countries - England, Scotland and Wales. Cornwall is another country on the island of Great Britain but it has no representation and is treated like a county of England.
When Scotland becomes an independent state it will leave the United Kingdom of Great Britain of Northern Ireland, though it will retain the same Monarch like many Commonwealth states. It just won't be a part of that politically-united "Kingdom" called the UK. The "United" part refers to political union i.e. a merging of borders and such. The countries that are "united" are reduced to province-like entities, "countries" only in a historical and barely legal sense.
Entities like the Overseas Territories are not part of the UK, but are British subjects.
The rump-state left over from the secession, commonly called the rUK or "rest of the UK" would no longer be the United Kingdom of Great Britain and Northern Ireland. "Great Britain" includes Scotland, after all. We will always be British for the reason that we are native to the island called Britain. It is a geographic term.
This means that the UK would have to become something like the "United Kingdom of England, Wales and Northern Ireland" and maybe Cornwall would arrive at a later date.