In 1916 the Irish revolution kicked off with the Easter Rising. It began with the reading of a document known as the Proclamation that proclaimed the establishment of the Irish Republic - a free and independent state. The full document was read by Patrick Pearse outside the General Post Office in Dublin on Easter Monday, 1916. Rebels then seized the Post Office and other buildings in the city. The Irish forces who did this were made up around 1,500 men and women of the Irish Citizen Army, The Irish Republican Brotherhood, the Irish Volunteers and Cumann na Mban. Collectively, from the eve of the Rising, they called themselves the Irish Republican Army or IRA. They proclaimed the Irish Republic and ran the flag of the proclaimed republic up the mast over the GPO. British forces numbering around 25,000 moved in on the Irish forces and after a week of fighting the Irish surrendered. The leaders were rounded up and shot by the British.

These actions led into the Irish War of Independence where the IRA found themselves in battle with the British. It also led to the political party Sinn Féin coming to the fore in Irish politics and contesting the general Election of 1918. Sinn Féin had not taken part in the Rising although many newspapers of the period reported that they did. Sinn Féin won 73 out of a possible 103 seats but refused to take their seats at Westminster – as they had promised the Irish electorate - and they set up an Irish government in Dublin in January 1919. They named the government Dáil Éireann and established a President as head of state. They then proclaimed the Irish Republic with an official Declaration of Independence [it applied to the whole of Ireland] and established the Irish Republican Army [IRA] as the legitimate and official army of the now proclaimed Irish Republic. The British immediately declared the Dáil to be illegal and sought to arrest the Sinn Féin members.. In the meanwhile, the Westminster Parliament had passed a bill in 1920 dividing the country into two and establishing a border with the new region of ‘Northern Ireland’ taking in 6 of the 9 Ulster counties. The Dáil refused to recognize this bill.

The British Army – augmented by the notorious Auxiliary force and Black and Tans - were now in a battle with the IRA which went on until a truce was declared in July 1921 and negotiations set for Dec of that year. The Treaty that emerged with the truce was a contested document within Ireland. The British, led by Lloyd George and Winston Churchill, refused to recognise the Irish Republic and offered a ‘Free State’ only. The Irish Republic had a president as head of state. The Free State meant that the British King was to be accepted as head of state. The Treaty applied only to the region outside of the new – some described as gerrymandered – cordoned off Northern Ireland. In Dublin Dáil Éireann voted to accept the Treaty but it was a divided vote. This led into the Irish Civil War as the IRA and Sinn Féin split on the issue. Many felt that they had fought for years – and lives had been given - for “the Republic as proclaimed by Pearse” as the narrative went. In the end the Anglo-Irish Treaty prevailed and Ireland was split into two. The Free State left the United Kingdom and Northern Ireland remained within the United Kingdom and under the Union flag. The Irish Free State would eventually break from the British Monarchy in 1937 - and elect a President - and from the British Commonwealth and become a Republic in 1948.

Edit: Further reading- Atlas of the Irish Revolution. Edited by John Crowley. Mike Murphy.

To expand slightly on /u/BrehonDubh's answer:

In October 1917 Éamon de Valera was elected President of Sinn Féin as a compromise candidate between the the republican wing and constitutionalists who were prepared to maintain a position within the British Empire. De Valera's proposal was the Sinn Féin would declare and achieve international recognition of the Irish Republic, after which the Irish people could freely choose their own form of government and relationship with Britain. This helped him to keep militant republicans like Cathal Brugha and Austin Stack onside along with those like Arthur Griffith who favored a duel monarchy such as had existed in the Austro-Hungarian Empire.

De Valera wasn't a doctrinaire republican and used it mainly as a tool to keep the militants on side. When negotiations began with Britain after the Truce on the 11th of July 1921 de Valera hoped they would except "Document No. 2", also known as External Association and the original question could be framed as "What's the difference between an Irish Free State and External Association?" This policy is very unclear and caused a great deal of confusion but what it boils down to is that Ireland would be "associated" with the Commonwealth for a number of matters such as war and political treaties and Ireland would recognise the King as the head of this Association.

De Valera felt this would satisfy the republicans by creating a republic in all but name and satisfy the moderates and the British government by maintaining a link with the Commonwealth. External Association was briefly discussed during the Anglo-Irish Treaty negotiations in London but was shot down by the British who insisted that the King would be the King of Ireland and would be given a position within the Constitution, including an oath of allegiance. Ireland's position in the Empire and the status of the King were left until late in the debates when the Irish team had used up what leverage they had on other concessions and were pushed into accepting the Oath and the King.

Cathal Brugha was supportive of External Association but not the Treaty that was signed. De Valera, like I said not a doctrinaire republican, was dragged into backing him and other extremists like Mary McSwiney when they came out strongly against it. He felt that with their support they could defeat the Treaty in Dáil Éireann and have it renegotiated more in line with his compromise. It is certain that had this happened the British would not have entertained External Association and may have used the rejection as a pretext to increase military operations in the country.

External Association could possibly have prevented the Civil War. While the IRA wanted nothing less than a full republic, they could have gotten behind a Treaty supported by the likes of Brugha. External Association also prefigured the evolution of the Commonwealth which now includes republics like India who have their own head of state but recognise the Queen as head of the Association. The Civil War is a lot more than just a fight over the oath of allegiance as well though. There had long been a divide between those who wanted to split fully from Britain and those who wanted to remain, many former Home Rulers and Southern Unionists would have been in the latter group. Those who opposed the Treaty believed those in favor were destroying the Republic and keeping in power the people they had spent the previous few years fighting against. Those who supported the Treaty looked at the other side as being delusional, thinking they could take on the might of the British army in a renewed war, refusing to accept a Treaty that gave Ireland much more power than it would have had under a Home Rule parliament.

The Civil War is a very complex topic, often shortened to "De Valera started it" and it will be interesting to see new perspectives into what caused it in the years ahead.