Regularly posters get flamed for calling the Republic of Ireland, Eire or Southern Ireland. This is seen by some as pandering to the British as the British Media insist on using the terms Eire or Southern Ireland, as a way of differentiating between the North (the part inside the UK) and the Republic. If you don't wish to offend use the term Republic of Ireland when refering to the 26 counties.
In a similar vein use Northern Ireland when refering to the part inside the UK as Ulster consists of nine counties, three of which are in the Republic.
The Constitution of Ireland (Bunreacht na hEireann) states in Article 4.
The name of the State is Eire, or in the English language, Ireland
The total island of Ireland has 32 counties, 26 are in the Republic of Ireland and the remaining 6 are in the North of Ireland. The island is historically divided into the four provinces of Leinster, Munster, Connacht and Ulster. The province of Ulster has 9 counties, of which 6 are in Northern Ireland.
Northern Ireland is part of the United Kingdom, and the remaining 26 counties make up the Republic of Ireland.
This is further confused by the fact that articles 2 & 3 in the constitution claim the North as part of Ireland.
The national territory consists of the whole of the island of Ireland, its islands and the territorial seas.
Pending the re-integration of the national territory, and without prejudice to the right of Parliament and Government established by the Constitution to exercise jurisdiction over the whole of that territory, the laws enacted by that Parliament shall have the like area and extent of Saorstat Eireann and the like extra-territorial effect.
More information can be found from various ftp and www sites given in section 11.