It really gets somewhat complicated but I'll try.
The group of Islands at the north west corner of Europe are the British Isles. "British' is a geographic term that applies to any person, language or culture that is associated with those islands. Those islands have three distinct political entities; the largest being the "United Kingdom of Great Britain and Northern Ireland" The second largest is the "Republic of Eire" this comprises of most of Ireland excluding the 6 counties of Ulster or Northern Ireland. The smallest is an island the "Isle of Man"
In between the United Kingdom and France are the Channel islands, a group of small islands, each with a measure of autonomy.
The largest political entity is the United Kingdom. The "Great" in Great Britain refers to the "Greater part of" So the United Kingdom is the greater part of the islands of Britain. The United Kingdom consists of 4 historic nations; England, Scotland, Wales and Cornwall. each with a distinct history and language. Those 4 nations send their representatives to the UK parliament and that government controls the military, makes treaties with foreign countries and does all the things that a sovereign state is expected to do. Scotland, Wales and Northern Ireland have some control over local affairs but the decisions regarding the Island are taken by the UK parliament.
So, to sum up. Anyone from the islands of Britain is British. That may seem obvious but it is a highly controversial political statement that large numbers of people will disagree with.
The people of England are English. The people of Scotland are Scots, their culture is Scottish. The people of Wales are Welsh and the people of Cornwall are Cornish.