Ever-so vaguely: "What do we, 'the British', do about 'Ireland' and 'the Irish'?" It is a "Question", because many languages use the same word for and do not make a real distinction between "question" and "problem". German (whose roots are most similar to those of English, Dutch/Flemish aside) and Mandarin Chinese (the world's most-spoken first language) are two such languages.
Many modern countries were once British protectorates, or administered by Britain as part of her Empire. Ireland is unique in actually having been an integrated part of the United Kingdom (of England/Wales, Scotland, and Ireland), in the same way Poland was once divided between and integrated into Prussia, Austria and Russia in the 18th century. Like Poland, proto-nationalism never quite died in spite of this due to Napoleon I's establishment of a semi-independent Grand Duchy of Poland and the sentiments of the literate elite who composed the vanguard of nationalism as we know it today in the 18th and 19th centuries.
Depending on how you consider it, the Irish Question might cover the whole of Irish-British relations from well before the incomplete Norman conquest during the The Middle Ages to the present. Usually, however, it refers to the period between 1801 and 1922, when Ireland was formally a part of the United Kingdom of Great Britain and Ireland. Wales had long been assimilated into English society, while Scotland entered into a mutual agreement (encouraged, according to Robbie Burns, by copious amounts of English gold paid to the Scottish negotiators) with England in 1707 to become the Kingdom of Great Britain, which suited the Lowland Scots quite well and the Highland Scots not at all, as they benefited little from the ensuing economic boom and then staged two failed rebellions to destroy the Union and preserve their power. The United Kingdom had the Kingdom of Ireland under its thumb for nearly a century by the time the Irish Parliament voted to join the Union and its members went to sit at the National Parliament in Westminster, like everyone else.
Given the nature of the "United Kingdom", it's natural to ask why there was not a Cornish Question, or a Welsh Question, or a Scottish Question, or even an English Question (Actually, there is sort of an English Question, as well as 2014's Scotland question). The answer is simple: they had very little sense of nationalism as we know it today and shared a broad non-Catholic allegiance with each other by the time of the Union of England and Scotland, both of Anglican and Calvinist leanings, respectively. Most Irishmen, on the other hand, were and are Roman Catholic. One can only wonder how different the fates of Britain and Ireland would have been had a certain English monarch not been so eager for a divorce.
As a sense of national identity developed, many Catholics became nationalists, seeking independence, or at least local autonomy. Scottish Protestant settlers formed the core of the segment of the Irish people who identified more with Britain, most of which lived around Ulster. That their sympathies would be such was deliberate, as they had been sent to settle for this very purpose; that they settled down in Ulster was more a question of its proximity to Scotland (Ironically, the earliest Scots did migrate from Ulster centuries ago). Likewise, Dublin had a long history of English immigration, and was a broadly Anglican area. Over time, the Protestant community in and around Dublin came to identify more with Irish culture than their Ulster counterparts (for example, Jonathan Swift, a Dublin-born Englishman, wrote plaudits to the Irish), whose more standoffish faith led them along much more firm religious and eventually nationalist lines.
Broadly, most of those who wished for Irish independence were Catholic, and equally broadly those who supported the union of Ireland with England and Scotland ("unionists") were Protestants. Exceptions abounded.note
The situation was soured by an undeniable streak of anti-Irish racism in Britain. Detailed more thoroughly on the page for The Irish Diaspora, the Irish were often ridiculed in newspapers and magazines, portrayed as shambling ape-like thugs in political cartoons (such as the image above), deemed unfit for self-government even by better-minded Britons, and often despised (and feared) for their poverty, propensity for violence, and— above all— their insistence on their Catholic faith. Since the sixteenth century, the Roman Catholic Church was frequently identified with foreign enemies seeking to invade and destroy British culture; against the backdrop of the Spanish Armada, the Gunpowder Plot, the Jacobite uprisings, and near-constant war with France, it is easy to see why British Catholics were almost always treated with suspicion. This view changed over the course of the 19th century, as anti-Irish rancor faded and by the late 1880s had largely died down.
On the other side of the Irish Sea, hostility was worsened by a truly horrific famine in Ireland in the late 1840s, which killed a million and a half people and forced another one-and-a-half-million to go elsewhere. Many Irish believed that it was caused by utter stupidity at best and deliberate malice at worst on the part of London, who preferred starvation in Ireland to chaos in Britain should the former stop exporting food to industrial towns (the famine could have been avoided altogether had there been a basic crop rotation system, or the Irish themselves grew multiple varieties of potato — both tried-and-true agricultural practice for centuries; this gets glossed over a lot but is important to know, because at the time most farmers were tenants who could not afford to feed their families any other way). A policy often pointed to is the Corn Law forbidding Irish farmers from growing this crop and thus competing with English ones. They then turned to potatoes, which failed, and the famine struck.
Until 1916, most Irish nationalists were not republican; most even envisioned an autonomous state which recognizes the sovereignty of the House of Saxe-Coburg and Gotha (later Windsor post-World War I), often citing Canada as a model. Both Queen Victoria and Edward VII were popular and received enthusiastic welcomes on visits to Ireland. Indeed, Victoria had a particular personal fondness for Ireland, often holidaying in Kerry. The full break only came later on when things had gotten worse.
There were many political attempts to reconcile Ireland into a Home Rule arrangement that would (like the current devolution of Scotland and Wales) keep Ireland in the UK. These were the efforts of the moderate nationalists of the Irish Parliamentary Party (or simply, "Home Rule Party") and Gladstone's Liberal Party, but they were systematically crippled by the joint efforts of the Unionists and the Tories. The first Bill was summarily shot down, the second attempt passed through the House of Commons but could not make it through the conservative stronghold of the House of Lords. Finally, the third attempt passed through both Houses and even received Royal Assent, but its implementation was not-at-all-co-incidentally delayed by the onset of World War I - destroying Germany and submerging domestic tensions (safe and humane working conditions, provision of living wages, women's suffrage, the future of Ireland) were the prime motivations for war among the hard-liners in the Liberal-lead Coalition Cabinet, though it was ultimately safe-guarding the post-Napoleonic order that Britain had helped established in 1815 (vis à vis the preservation of Belgian independence) that won over the undecided ministers and the voting public. In 1916, during the war, a couple of hundred radicals staged an armed uprising on Easter Monday in Dublin, declared the free and independent Irish Republic, and were almost all killed or imprisoned by the British Army. The public took a dim view of the rising initially, with reactions ranging from bewilderment to outright contempt, considering it something of a betrayal, especially as many Irishmen were then serving with the British Army in France. However, public opinion then changed to general shock and outspoken disapproval at the brutal (by British standards) treatment of the dozens of captured rebels, the trial by military courts and subsequent execution of many of the surviving leaders, and a Draconian policy of repression to cut down on further would-be-martyrs. It became increasingly clear to the Irish people that to the British the "rights of small nations" they were being asked to fight for in Belgium did not include themselves.
The continued Cultural Revival of neo-Gaelic culture, and general war-wariness, increased popular support for revolutionary nationalist groups. This was dramatically exacerbated in 1918 when - with the country's manpower reserves exhausted - the possibility of extending the 1916 Conscription Act to Ireland was contemplated by the Cabinet. This was staunchly opposed by many within Ireland, and both the Unionist and Nationalist parties condemned the proposalnote .
Then came the end of the war and the 1918 general election, which was contested by the newly-relevant Sinn Féin ("We ourselves"), running on a platform of abstention from Westminster and the establishment of an independent Irish Republic. They took the election in a massive landslide, taking all but two constituencies in the country (outside of the Unionist stronghold of East Ulster, naturally, where they won nothing) - helped by the fact that Westminster had just granted suffrage to all men over 21 and women over 30, giving the nationalist majority the electoral advantage for the first time. They also elected Countess Constance Markievicz, the first woman elected as MP to Westminster, though of course she would never sit there. Soon afterwards the Irish Parliamentary Party (moderate nationalists who were the principal drivers behind Home Rule since the 1870s) would dwindle to nothing, replaced by Sinn Féin as the main political force in Ireland. Sinn Féin formed the First Dáil Éireann in January of 1919 and began the long task of setting up the new architecture of state. That same day, however, the War of Independence began. At first confined to small local raids on police and army barracks, the scope of the conflict gradually increased as the British government took a more aggressive stance against the IRA. Assassinations of officials and acts of violence against and murders of 'pro-English' citizens increased, and resulted in increasingly heavy-handed government repression with many hundreds of Irish citizens being beaten, arrested, killed extra-judicially, sentenced to prison and hanged. The decision to recruit WWI veterans to serve as armed policemen (the 'Black and Tans' and 'Auxies') in particular proved to be a bad move, as they hired people trained and indoctrinated to exact vengeance upon a hated enemy for use in a delicate domestic political situation that called for a great deal of understanding and self-restraint.note Even those 'Black and Tans' who did not have a psychological dependency on warfare or sign up for the opportunity to kill people, usually with complete impunity, had received fundamentally different training and conditioning than for what their new role required there.
Hundreds died in the cycle of violence and (increasingly lethal) repression that followed until the Irish War of Independence was concluded with the Anglo-Irish Treaty of 1921. The Irish Free State was acknowledged independent, while Northern Ireland remained British. Following this there was some severe disagreement within Sinn Féin with the terms of the Treaty partitioning the country, and there followed a brief and exceedingly bitter civil war that sadly killed more than the War of Independence. The civil war still affects Irish politics to this day—the two major modern parties, Fianna Fáil and Fine Gael, are descendants of the Anti- and Pro-Treaty Sinn Féin factions, respectively. The Civil War is the Elephant in the Living Room in Irish culture, and doesn't get mentioned much for fear of causing offence; notable exceptions are Sean O'Casey's play Juno and the Paycock and the recent Ken Loach Film The Wind That Shakes the Barley. The Irish Free State eventually went on to become the current Republic of Ireland by unilateral declaration in 1949, subsequently recognized by the UK (but they didn't change the license plates until 1987). For Northern Ireland see The Troubles.
This area of history has a lot of disagreement; suffice to say everyone has his or her own view on the subject.
The conventional, conciliatory view is that British people should bear in mind the people who were imprisoned or died as a result of a couple of centuries of repression, neglect and mismanagement by British governments, and that perhaps if a nation has five separate armed rebellions for independence in a hundred years and two separate political parties dedicated solely to that cause, you should maybe take the hint. The Irish (a nation with a long and noble history of grudge-holding) in turn promise to stop holding grudges over grievances that can, in some cases, be four hundred years old. British-Irish relations are, after all, important to both countries - and neither a culture of historical ignorance nor one of victimization and grudge-holding is healthy - or, incidentally, a recipe for future harmony between two very close geographical, economic, and cultural neighbors. Of course, in truth, almost everybody under the age of fifty from either nation (outside professional politics and outside Northern Ireland, which has its own problems) takes no particular heed of an increasingly-distant history, or at least don't take it personally.
A significant symbolic step towards reconciliation was made in 2011, with Queen Elizabeth II making a state visit to Ireland — the first visit from a British monarch since King George V's visit in 1911, and the first ever since Ireland gained independence. Although subject to criticism and protest from numerous quarters, the visit was widely regarded as a success and the Queen widely praised both in Britain and in Ireland, particularly for a speech delivered at a state dinner in which she both acknowledged the regrettable past between the two nations while asserting how much closer, richer, and happier the British and Irish peoples have gotten since then.note
Do not confuse with "The Jewish Question".