So this is an enormous question, and it’s going to get an enormous wall of [structured] text in return. I’ll start off with a bit of context on the history of Ireland, before moving into the Easter Rising and the Anglo-Irish War. After that, I’ll look at the Troubles in the latter half of the 20th century. In between the paragraphs on those sections, I’ll put some summations that will hopefully bring it all together in a clearer fashion. I’ll try chuck in a tl;dr at the end, but I can’t promise that that won’t be huge as well.

Ireland was a conquered nation – invaded by England nearly a nine hundred years ago and ruled ever since. About halfway through that period, in the 15th and 16th centuries, Irish land was confiscated by the English rulers, and given to Protestant English and Scottish gentry. These became plantations, and in effect, the colonisation of Ireland by England. Throughout the years there were numerous rebellions by sections of the Irish population against English rule. These rebellions were ruthlessly shut down, and the leaders executed. The Irish public was also severely punished for this. One of the key figures in this was Oliver Cromwell, who is noted for his harsh treatment of the Irish, and ordering and being involved in the commission of what today we would term war crimes. In 1798, the United Irishmen, with aid from France, rebelled against the Crown. This rebellion was put down too, and in 1800, the parliaments of England and Ireland passed the Act of Union, merging the Kingdom of Ireland with the Kingdom of Great Britain to create the United Kingdom of Great Britain and Ireland. At the start of the 20th century then, Ireland had been ruled by England for around 700 years, and had been a forced member of the Union for 100. In the later years of the 19th century, a movement advocating home rule had emerged in Ireland, calling for Ireland to have self-rule, while remaining within the Union. While remaining within the Union wasn’t necessarily a goal, it was a compromise, and self-rule(Home Rule) was a first step. A bill granting Home Rule to Ireland was passed in the early years of the twentieth century, but there were two problems with it: firstly, Ulster, the northernmost province of Ireland, was a majority Protestant region of Ireland, and as such, were opposed to the dissolution of the Union. Secondly, the bill was put on hold in 1914 when war broke out in Europe, with an Act passed that meant it would not come into effect until after the war finished – in other words, at an unknown point in the future, Ireland would receive it. The Ulster Unionists created militia in the North, raising substantial numbers of men. In response, the Irish Volunteers were formed in the South, protecting the idea of Home Rule. With the outbreak of war, many of these Irishmen served as part of Irish regiments in the British Army. Those that stayed behind were against Irish involvement in the war, and on Easter Monday 1916, they, and other nationalists, seized buildings in Dublin and declared an Independent Ireland. Their numbers were small, and after six days, the rising was subdued, and sixteen participants were executed and many more imprisoned.

The Easter Rising was a curious event: it went against the majority of Irish opinion. Home Rule was in the works, it was guaranteed. Over a 150,000 of the Irish Volunteers had gone off to fight for Britain in the war, because they believed that Home Rule would be delivered. In contrast, this small group of rebels left behind had gone past Home Rule, and straight for independence. And they had done it violently, seizing buildings by force, and fighting the British Army on the streets of Dublin. This was done after a century of parliamentary manoeuvring towards Home Rule, and they threatened it all. The executions, the high death count(particularly among civilians), and the long timespan of the First World War all contributed to a growing support for the Irish Republic, and a diminishing tolerance of Britain.

At the United Kingdom general election in December of 1918, Sinn Fein, a nationalist political party won 70%(about 73 out of 105) of the seats allocated to Ireland in Westminster. The party had never won seats before, indicating a significant swing in public opinion towards the British. Their members refused to sit in Westminster, instead setting up an Irish parliament and government in Dublin, and re-declaring independence. This new parliament was not recognised by the British, and was in fact declared criminal in early 1919 – after this, they only met in secret. This outlawing of the Irish parliament is cited by some historians as the start of war, and from this point forward, the British presence in Ireland was at war until the truce in 1921.

So to summarise this first part of your question, there were several factors that led to the Irish War of Independence. Firstly was the English conquest and subsequent colonisation of Ireland, which had been cruelly enforced, and had arguably contributed to the enormous death toll of the Famine in the 19th century(see other /r/AskHistorians posts on whether the Famine was a genocide). Secondly, the Easter Rising of 1916 had resulted in some of the worst violence Ireland had seen in a century, and the executions were reminiscent of the treatment of the Irish after earlier rebellions. The martyrdom of the leaders was particularly effective in mobilising anti-British sentiment. The Sinn Fein party’s success in the 1918 general election is the third factor, and was representative of a change in Irish political opinion, and the subsequent withdrawal from Westminster and establishment of an Irish parliament stimulated support for the independence movement. The last factor was the British condemnation of that parliament, and the implicit condemnation of independence for Ireland.

I'm taking a break, but I'll come back and do a post on the Troubles later, e even though it really should be a different post.

EDIT: Word formatting sucks when copying to Reddit.

As others have mentioned this is a very complicated issue. Your have really asked two connected but different conflicts. The Irish War of Independence occurred between 1919 and 1921. The Northern Ireland ‘Troubles’ of 1969-98 occurred in a different context. I apologise for the length of this post but I think it needs a detailed answer.

 Looking first at the War of Independence, I’ll use the old fashioned long and short-term causes approach. The long-term cause was the failure of the British state to reconcile Irish demands for self government with British desires to preserve the UK.

During the second half of the 19th Century, Irish nationalism developed into its modern form supporting on self-government, a cultural revival of traditional language, customs and sports, land reform and the social and economic advancement of the Catholic population who formed its main supporters. British policy was to grant substantial reform of land and religious laws and to tolerate cultural revival. Any expression of violent resistance was met with force, however. This left the issue of self-government. British Liberals were prepared to allow Ireland to be self-governing in domestic matters (‘Home Rule’). The strategy of mainstream, constitutional Irish nationalism between 1870 and 1918 was to gain Home Rule. More radical nationalist and republicans wanted a completely independent republic and many of these favoured the use of violence. The British Conservative party opposed Home Rule and allied itself with Irish unionists. Home Rule was finally passed in 1914 but was not put into effect because of the First World war.

This brings us to the short term causes. The passage of Home Rule radicalised both sides who formed militias (the Irish Volunteers and the Ulster Volunteer Force) to enforce/defeat Home Rule should it happen. At the outbreak of WWI Ireland was on the brink of civil war. The outbreak of war split Irish nationalism with the majority of the Irish Volunteers supporting the war effort. The war seriously diminished the popularity of the Home Rule position for several reasons: 1) as the conflict dragged on it seemed that Home Rule would not be enacted, 2) a republican rising in 1916 was brutally defeated and large numbers of people who had not been involved in violence were arrested, 3) the fear that conscription would be extended to Ireland radicalised the population leading to mass protests in 1918, 4) the expansion of the franchise in 1918 allowed younger and working class men a vote. Through 1917 the republican Sinn Féin party won several by-elections and in the 1918 General Election it won 73 of the 101 Irish seats. It refused to take its seats in Westminster and formed a separatist parliament Dáil Éireann. Sinn Féin had not campaigned on a platform of violent revolution but when the Irish Republican Army (which contained many of the same people) began to attack police and army barracks the Dail endorsed the campaign.

The IRA fought a guerrilla campaign until 1921 when a truce was declared. In 1920 Britain introduced Home Rule was introduced while partitioning Ireland into a nationalist South and unionist North. Sinn Fein won the resultant southern election and refused to recognise Home Rule, continuing its claim to a Republic. Ulster Unionists dominated the Northern Ireland parliament and set about safeguarding their position by removing nationalists from positions of influence. Therefore partition was already a fact on the ground by mid-1921.

 British and Irish delegations negotiated a Treaty in December 1921. This gave the South the title of the Irish Free State and the status of a Dominion (such as Canada or Australia) The Dail and IRA split over the treaty and the resulting civil war was won by the pro-Treaty side. Meanwhile IRA resistance to the Northern Irish government was crushed by about 1924.

Moving on to the question about violence after 1923, The IRA continued at varying levels of activity in the North and South. It launched significant offensives in 1939 and 1956 which were defeated in both states. In spite of this the levels of violence in both states was low between 1923 and 1966. 

Northern Ireland did have a significant level of discrimination against Catholics (about 35-40% of the population) in politics, employment and public services (especially housing). There were no Jim Crow style laws but Catholics were considerably worse off and there was extensive gerrymandering of some electoral areas. This led to a civil rights campaign in the mid-1960s which destabilised the Unionist government. On the one hand the government came under British pressure to make concessions but it faced a backlash from reactionary unionists led by Ian Paisley among others who opposed all reforms. Loyalist paramilitary groups reorganised and were responsible for bombings in 1966.

Sectarian violence became serious in 1968-69 and British troops were deployed. The IRA split at this time and the more radical Provisional IRA began its offensive in 1970 which continued until the ceasefire of 1994 (it returned to violence from 1996-97). During this time it adopted a combination of guerrilla and terrorist tactics. Its aims were to topple the Northern Irish government, then to force the British to withdraw and set up a united Ireland under a new constitution. Loyalist paramilitaries focused their attacks on the Catholic communities which they saw as supporters of the IRA. This created a cycle of atrocity where tit-for-tat attacks and counter-attacks on Catholics and Protestant which gave the conflict its sectarian dimension. The British used a mixture of the army, special forces (the SAS) and the police. There are also legitimate reasons to believe that loyalist paramilitaries had collaborators within the police and intelligence services. The extent to which British intelligence had infiltrated republican and loyalist groups is still emerging.

 The Northern Irish government introduced internment without trial in 1971 to in an attempt to stop the IRA. It had been used successfully in the 1920s, 1940s and 1950s but failed badly in 1971; it was directed solely against nationalists and many of the wrong people were arrested. It only radicalised the nationalist population especially after an anti-Internment march in Derry was fired on by British paratroopers killing 13 people, an event known as ‘Bloody Sunday’. Soon afterwards, the British government took over the direct running of Northern Ireland and dissolved the local parliament. 

Mainstream politics did continue and the majority of the electorate consistently voted for parties which condemned violence. Paradoxically this may have actually lengthened the conflict as the paramilitaries were not vulnerable to public opinion outside their immediate localities. It was only in the 1980s that the IRA formulated a coherent electoral strategy and its political wing Sinn Fein did not become the largest nationalist party until after the peace.

The IRA was not especially large but was reorganised after the mid-1970s into a cell structure and it was very well armed due to arms shipments from Libya in the 1980s. It was therefore quite resilient. However it badly miscalculated the endurance of the British government. Northern Ireland never became a significant issue with the British public at election time and the IRA could not inflict a military defeat on the British state. We are still unclear about how exactly (or how fully) the IRA transitioned to purely democratic politics but it seems clear that its leadership (especially Gerry Adams and Martin McGuiness) were looking for a way out of the military stalemate. 

While the modern IRA believed themselves to be the direct descendants of the IRA of 1919-21, this would be disputes by many non-violent nationalists and republicans. Making comparisons has proven to be difficult as it is so politically contentious. It is also somewhat dangerous to see the Northern Irish troubles as an inevitable outcome of partition rather than as a conflict that occurred due to numerous failures in the period 1921-68 to deal positively with the serious problems in Northern Ireland

. Sources: CAIN website on the Northern Ireland Troubles: http://cain.ulst.ac.uk/index.html

Marie Coleman, The Irish Revolution 1916-1923 (2014) Richard English, Armed Struggle: the history of the IRA (2008) Richard English, Irish Freedom: the history of nationalism in Ireland (2008) Brian Hanley, The IRA 1926-36 (2002) David McKittrick & David McVea Making sense of the Troubles (2000). Peter Rose, How the troubles came to Northern Ireland (1999) Charles Townshend, The Republic: The Fight for Irish Independence, 1918-1923 (2013)