There's the United Kingdom of Great Britain and Northern Ireland. Great Britain is made up of Scotland, England and Wales. As it's a Kingdom, it's ruled by a Monarch - Queen Elizabeth II. As a result of her ancestors having their heads cut off or getting chased out of the country, the Queen has limited powers. The country doen't have a constitution, and any parliament can set whatever laws it wants.
It has a Parliament ( Congress) and the effective Head of Government is the Prime Minister who leads whichever Party in Parliament is the bigger. Just like the states, there are two Houses of Parliament - the House of Lords ( like the Senate, but not elected, they are now mostly old politicians) and the House of Commons (like the House of Representatives, elected, and the main seat of power).
The Republic of Ireland is a Republic on the same lines as the States, with a President. Interestingly, although the President in the Republic has similar powers to the US President, there is a custom that they let the Dail (Irish parliament) make the running.
Here in Northern Ireland, there is a continuing dispute over whether we should be ruled from London or Dublin. I'm not going to go into that as it's too painful and convoluted. We've just been through thirty years of murder and mayhem fighting over it. I'm just glad to have peace, for the sake of the kids.
That's probably too much information !
"What's so funny 'bout peace, love and understandin' ." Elvis Costello