The Troubles is a term used across the United Kingdom and the Republic of Ireland to refer to the conflict between Irish Republicans/nationalists, the British government, and Ulster unionists/loyalists for control over the six counties of Ulster, Northern Ireland, that remained under British sovereignty after the partition of Ireland in 1921.
From the late 1960s through the late 1990s, the British Army, various Protestant paramilitary organisations, the Provisional Irish Republican Army (IRA, aka the "Provos") and smaller Irish nationalist grouplets struggled for control of Northern Ireland. This conflict is a continuation of the long struggle for Irish independence from British rule that was expressed in religious terms only with the Protestant Reformation. Before that it was unadorned imperialism.
During the early/mid 12th century, Ireland was a series of petty kingdoms and chiefdoms . One of these petty Kings, Diarmuid McMurrough of Leinster, invited a Norman baron (Strongbow) over to Ireland to help him retake his petty kingdom in 1169. There were only a few hundred Normans involved, but since their armour and weapons made them seem like space aliens compared to the rather primitive Gaelic tribesmen, they pretty much kicked ass. Henry II, scared that a new Norman Kingdom would emerge in his backdoor (Strongbow was made heir to the Kingdom of Leinster when he married McMurrough's daughter), quickly sailed over the Irish Sea and subdued the Gaelic chieftains, demanding their fealty and pretty much controlled Ireland by 1172. Ireland was semi-independent for the next couple of hundred years. Most of its great feudal barons and magnates had Norman/English roots, but eventually they became culturally Irish (commonly referred to as 'the Old English').
Ireland's history changed irrevocably with the reign of Henry VIII. His break with the Pope created strife as the staunchly Catholic Irish refused to accept the new Church of England. Henry unsuccessfully attempted to foist it on them, and his daughter Elizabeth I (who had to fight coup and invasion attempts from Catholic individuals and organizations) did so as well. Finally, after numerous rebellions, in the first decade of the 1600s settlers were sent from Scotland and England to provide a more loyal, Protestant population, particularly in Ulster. This was stepped up by Oliver Cromwell during and after the English Civil War, in which the Irish had supported the losing side of King Charles I in another failed rebellion. Catholics were stripped of many rights, including to vote and hold public office, while most land was given to English noblemen.
Rebellion sputtered into the next century, ending in the Rising of 1798, supported by the French. The last vestige of Irish autonomy was ended as punishment for this, with their parliament abolished, becoming formally united to the Kingdom of Great Britain in 1801 to form the United Kingdom of Great Britain and Ireland, the countries of which were England, Wales, Scotland, and Ireland. Soon afterward, calls for Irish secession and independence from the Union came frequently from some of the Catholic community in Ireland. Home Rule and disestablishment of the Church of Ireland was an issue later that century under Liberal Prime Minister William Gladstone. The Liberal party split over the question of Union or home rule with the Liberal Unionists mostly absorbed by the Tories which would ultimately lead to the demise of the Liberal Party as a major political force.
The first two decades of the 20th century saw an increase in campaigns for Irish independence—the Easter Rising of 1916 in Dublin was a notable event in the campaign against British rule. Plans for Irish Home Rule were delayed by the First World War, but the Anglo-Irish War afterward caused Ireland to be partitioned in 1921 into the largely Catholic nationalist Southern Ireland, which in 1922 became a British Dominion known as the Irish Free State, later becoming the independent Republic of Ireland in 1949, and Northern Ireland, which has since remained a constituent country of the United Kingdom. Since the division of Ireland, a campaign has existed among some of the Northern Irish community for the secession of Northern Ireland from the United Kingdom to rejoin the rest of the island as a Republic. This was rekindled from 1969-1998 with years of violence before peace accords were made.
Northern Ireland is religiously divided between Catholics and Protestants, with a significant minority of Catholics being traditionally nationalist and wanting a united Ireland, while almost all Protestants and a plurality (47%) of Catholics are unionist and consider themselves British and/or Northern Irish, seeking to maintain Northern Ireland's constituent country status within the United Kingdom (though it should be noted that many pro-union Catholics are not unionist in a traditional sense, and merely support Northern Ireland's position within the UK for fiscal reasons). Much of the Northern Irish unionist community is descended from Scottish (and some northern English) immigrants to the island in the centuries preceding the Union, hence the inherently Protestant belief system held by that community. There is also a (very) small proportion of the Northern Irish population who seek to form a breakaway Northern Ireland independent from the Republic of Ireland and the United Kingdom.
In the wake of the Brexit vote where Northern Ireland as a whole voted slightly in favor of "remain" (though the UK as a whole voted ~ 52-48 in favor of "leave") sectarian fault lines are feared to break open once more. Over 80% of Catholics voted remain while only 40% of Protestants did so. The DUP (now in a coalition with Theresa May's Conservatives in Westminster) is an openly Eurosceptic party and welcomes Brexit whereas Sinn Fein leader Gerry Adams has called for the "border poll" provided for in the Good Friday Agreement to be held as soon as possible in order to keep Northern Ireland in the EU. The British government has rejected both a veto right for devolved assemblies (like the Scottish or Northern Irish ones) on Brexit or a special status of any kind for parts of the UK or its dependencies, be they Scotland, Northern Ireland or Gibraltar thus making a "reverse Greenland" all but impossible.