Think of it this way…
Great Britain is the island that contains England, Scotland and Wales. Those three are separate historical nations that are located on that island. England, Scotland, Wales, and Northern Ireland are subordinate polities within the UK, sort of similar to US States, but it’s a very imperfect analogy. In many ways, US States have more sovereignty than the constituent nations of the UK. But there is more identity involved being with English, Scottish, Welsh, and Northern Irish than there is with being from most US States (Texas, parts of Louisiana (Acadiana) and Hawaii may be exceptions), as that’s not only a political identity, but an ethnic identity. Someone is English, and may be a resident and citizen of the UK, for example. I once dated a girl who grew up near Southampton, but who considered herself Scottish, because her parents were Scots, and while she identifed as British and a citizen of the UK, she didn’t identify as English, no matter how classically English her accent was, and the fact that she had grown up in Hampshire.
The United Kingdom is a political entity that comprises the nations of England, Scotland, Wales, and Northern Island (on the island of Ireland). The UK is the primary international political entity where foreign policy is set, the military is employed, and so forth. The UK is what everyone else thinks of as the “national government”- it’s what is represented in the UN, it’s what is represented in NATO, it’s what is represented in the EU.
The Commonwealth is a entity that isn’t a unified political entity any more, but is a sort of cultural union that is made up of former nations of the British Empire, but after a certain point. For example, India is part of the Commonwealth, but the US is not.
The European Union (EU) is a supranational union of which the UK was once a member. Brexit ended that.
The Crown is everything that is ostensibly ruled by the monarch of the UK- right now, King Charles III. This is the UK and the Crown Dependencies. Part of the uniqueness of the Crown Dependencies is that they’re territories that the sovereign rules in his capacity as the King because of his historical title of Duke of Normandy. Meaning that when William the Conqueror became King of England, he also retained overlordship over the Crown Dependencies.
The British Overseas Territories are places outside of Great Britain and Ireland that the goverment of the UK has authority, although they don’t have Parliamentary representation from what I understand.
Parliament is the UK’s elected legislative body, with the Prime Minister acting as Head of Government, while the King (or Queen) acts as Head of State. This is a weird distinction to Americans, as the President fulfills both roles in the US.
The Crown is a sort of vague, catch-all term for the institution of the monarchy and what it represents, as personified by the King or Queen. So The Crown represents the legal authority of the courts, the military, etc… as personified by the King, for example. It’s an amorphous idea, and one that isn’t easy to explain, but that isn’t too hard to understand.