Okay. So, here's a helpful primer on British and Irish stuff.
You have a collection of islands, commonly called the British Isles. This archipelago's name is sometimes objected to by the Irish, who object to it all being called British, but I'm going to use that term for convenience.
The British Isles is comprised of two major islands and lots of tiny little islands. The two big islands are the island of Great Britain, which is to the east, and the island of Ireland, which is to the west.
On the island of Great Britain, there exists England, Scotland, and Wales. On the island of Ireland, there exists Northern Ireland and the Republic of Ireland (whose formal name is just "Ireland," but which I'll call the Republic of Ireland to avoid confusion with the island).
England, Wales, Scotland, and Northern Ireland are all part of a single sovereign state called the United Kingdom of Great Britain and Northern Ireland. The Republic of Ireland used to be part of the United Kingdom, but they rebelled because, hey, turns out people don't like being part of a country without their consent. However, the six northeastern counties in Ireland stayed, forming Northern Ireland.
England has the largest population of the political subdivisions of the U.K., and its capital, London, is the capital of the U.K. As a result, many argue that England dominates the U.K. However, England is not the same thing as the United Kingdom.
The terms "U.K.," "United Kingdom," and "Great Britain" are often used as synonyms for the same thing -- the United Kingdom of Great Britain and Northern Ireland.
Something that's English is exclusively from or relating to England. Something that's Scottish deals with Scotland. Something that's Welsh deals with Wales. Something that's Northern Irish deals with Northern Ireland. Something that's British deals in some way with the whole United Kingdom, and therefore may also be English, Scottish, Welsh, Northern Irish, or all of the above. For instance, Russell T. Davies, creator of Torchwood, is simultaneously a Welshman and a Briton. He is not English, even though he is British.
England, Scotland, and Wales all used to be separate states: The Kingdom of England, the Kingdom of Scotland, and the Principality of Wales. Once upon a time, the Kingdom of England conquered the Principality of Wales, leaving only the Kingdom of England and the Kingdom of Scotland.
Meanwhile, on the island of Ireland, there existed the Kingdom of Ireland. The Kingdom of Ireland was in theory independent, but in reality was under England's heel.
Throughout most of their history, the Kingdom of England and the Kingdom of Scotland hated each other and went to war all the time. However, after the death of English Queen Elizabeth I, the King of Scotland inherited the English throne. England and Scotland were still independent, but they shared the same kings for over a hundred years. (There was a civil war and an interregnum, but they don't really matter for this discussion.)
In 1707, the Kingdoms of England and Scotland passed the Acts of Union 1707; they abolished themselves and formed the new Kingdom of Great Britain, which encompassed England, Scotland, and Wales. (Wales had been considered part of the Kingdom of England at the time; today it's recognized as its own separate thing in most contexts.) The Kingdom of Great Britain and the Kingdom of Ireland existed separately, though Ireland remained under the new Kingdom of Great Britain's heel. It was from the Kingdom of Great Britain that the United States declared its independence in 1776.
Finally, the Kingdom of Great Britain and the Kingdom of Ireland passed the Acts of Union 1800, which abolished the Kingdom of Great Britain and the Kingdom of Ireland and created in their place the new United Kingdom of Great Britain and Ireland.
The Irish were part of the U.K. for about a hundred and twenty years. Then, in the 1920s, they rebelled, 'cos, hey, guess what, they wanted to run their own damn country for once. Most of the Irish were Catholic and wanted an Irish republic (i.e., they were Republicans), but a majority in the six northeastern counties were Protestant and wanted to stay in the U.K. (i.e., they were Unionists). Finally, after the war ended, most of Ireland was allowed to form the Irish Free State, which was a new, sovereign state that shared a monarch with the United Kingdom in the same way that Canada still does, while the six counties in the northeast stayed in the U.K., forming Northern Ireland. After a few years, the Irish Free State abolished itself and replaced itself with the Republic of Ireland (formally just named Ireland).
There was a lot of fighting between Republicans and Unionists in Northern Ireland for about thirty years, called the Troubles, but the Troubles have mostly subsided. Northern Ireland remains part of the U.K., but no one knows what the future holds.
So, now, today, the United Kingdom of Great Britain and Northern Ireland (consisting of England, Wales, Scotland, and Northern Ireland on the island of Great Britain and in the northeastern part of the island of Ireland) gets to live happily alongside the Republic of Ireland (located on the island of Ireland).
Make sense?