Basically to understand how this state of affairs you have to understand the longer term history of English and later British imperialism in Ireland. I did recently write another very large body of text dealing with this long history. This process of conquest and colonisation began in the 12th century and was finalised in the 16th and 17th centuries. 

For the purposes of how there came to be a group of people in North-east Ulster utterly opposed to Irish rule, the most relevant development was the Plantation of Ulster in the early 17th century. I deal with this more comprehensively in the answer linked above so I’ll resist rambling on, but essentially this was a complex piece of social engineering in which land confiscated from Irish lords would be granted to British landowners. In turn, loyal Protestant settlers would be ‘planted’ on these lands as tenants in place of the 'barbarous' Irish natives. Attempts at plantation and colonial settlement in Ireland were not anything new necessarily, but the scale of the Ulster Plantation was much more comprehensive than anything seen previously. 

Another key factor is that Ulster’s proximity to Scotland meant that there was also a large influx of 'unofficial' settlers who came across too. Particularly in the decades that followed. For instance, the 1690s saw another pronounced wave of settlement into these plantation areas. 

There were Protestants all over Ireland of course, but as a consequence of the plantation only in North-East Ulster was there any kind of substantial majority. 

I am generalising here for the sake of brevity and simplicity, but basically the rest of the country saw a Protestant Ascendancy in the form of an elite landlord class with a subservient Catholic underclass. It was primarily this area of Ulster which experienced sustained Protestant settlement at a lower social scale and which therefore came to make up a greater proportion of the population. Not to say there was no Protestants elsewhere. The events of the "Glorious Revolution" and the Williamite Wars in Ireland in the 1690s would serve to confirm absolutely Protestant domination in Ireland. 

Over a century later, in 1800 Ireland was then brought into the United Kingdom through the Act of Union (a response to the 1798 rebellion). Of course Ireland had long been an English/British colony, but it was not directly politically integrated until then. As the nineteenth century continued the old Protestant Ascendancy began to be dismantled through a combination of agrarian agitation and political reform. Oppressive laws against Catholics were repealed, and they slowly began to have a more active role to play in political life as the century unfolded. Essentially this left this area of North-East Ulster as the only part of Ireland still under any kind of real Protestant control. In a number of these Ulster counties Protestants were a majority of the population. 

In the latter half of the nineteenth century we begin to see demands for something called  "Home Rule" from constitutional nationalists in Ireland. Certainly not for an independent Irish Republic, not even the Repeal of the Union. Home Rule was an incredibly limited form of devolution within the United Kingdom. There were some radical groups who did want to go further, but these were a firm minority at this stage. Nonetheless, for those Protestants in Ulster - known as Unionists - even Home Rule was a step too far. Descendants of the the seventeenth century planters they felt (and in many cases still feel) themselves and their identity to be deeply British. 

With the rise of the Home Rule movement the unionists feared any kind of political separation. They were a majority within their corner of Ulster, but would be a minority within Ireland as a whole. There was also a religious component too. Ulster Protestants feared that Home Rule would essentially be “Rome Rule”, to use the language of their own propaganda. 

As Home Rule became increasingly likely with the introduction of the Government of Ireland Act in 1912 (known as the Third Home Rule Bill), thousands of Ulster Unionists signed a document known as the Ulster Covenant. Ominously they vowed to use:

"all means which may be found necessary to defeat the present conspiracy to set up a Home Rule Parliament in Ireland"

Indeed that same year they also founded the Ulster Volunteer Force (UVF), a paramilitary group (though not the same as the modern UVF) and threatened civil war. The Irish Volunteers were set up in response. Initially it was hoped that the Ulster Unionists could stop Home Rule at all, but when it became apparent that this wasn’t possible they attempted to have to simply have Ulster excluded from the bill. It wasn’t only in Ulster that Unionists were to be found, but it was only Ulster which had any kind of serious political power to prevent it, for the reasons noted above. 

Only WWI stopped things breaking into civil war in all likelihood. However, things escalated again after 1916 when an Irish Republic was declared in the Easter Rising. When the Irish War of Independence broke out in 1919 it was clear that things had moved well past Home Rule. Sinn Fein had won an election running on the promise of an Irish Republic. Suddenly Home Rule didn’t seem so bad. So with war raging in the southern part of the country, a new bill was passed in 1921 which set up two different Home Rule parliaments - one in Dublin and one in Belfast, in the newly created state of Northern Ireland. This was the partition of Ireland. Ironically the one part of Ireland that didn’t want Home Rule was the part that actually got it!

Northern Ireland consists of six of the nine counties of Ulster. Going by areas with a Protestant Unionist majority, it should only have been four counties. But unionists feared that such a small area would not besustainable, while equally fearing that partition for the whole of Ulster would leave too large a Catholic nationalist population. So they agreed to a six county state, leaving a significant Catholic nationalist minority in many areas and counties but still with a dominant unionist majority in the whole state. 

Preoccupied with the war in the south and their aim of independence the original IRA and Sinn Fein didn't, or couldn't, do much to stop partition. The new Northern Ireland state was thus dominated by a unionist government determined to secure its position. After the Irish War of Independence, the rest of the island obtained 'Free State' status, which was a dominion like Canada or South Africa (so mostly independent but still technically within the British Empire), and then in the 1940s they declared themselves a full independent republic.

Thus today we have a separate Republic of Ireland, and Northern Ireland which remains part of the United Kingdom.