A United Ireland is what is desired by many Irish Nationalists or Republicans who desire to see Northern Ireland become part of the Republic of Ireland.
Elections were held for the Nationalist Parliament in Dublin in May 1921 in which Sinn Féin (under Eamon de Valera) took 124 seats with the remaining 4 being taken by Unionist candidates. Sinn Féin refused to recognise the Parliament and instead continued to meet in Dáil Éireann. The 4 Unionists were the only ones who attended the new Parliament. The IRA, under Michael Collins, continued to fight on for more independence, and made regular attacks on Protestants in Northern Ireland too. Finally stalemate was reached and a truce was signed between the IRA and the British on 11 July 1921. After 4 months of negotiations a treaty was hammered out which Michael Collins signed on behalf of the IRA. This lead to the Partition of Ireland.
Since that time the desire of Irish nationalists for a United Ireland has been resisted by Ulster Unionists.