The official name of the Country comprised of the 26 counties that form a Republic is "Ireland". It is also sometimes referred to as the Republic of Ireland. There is no Country called Southern Ireland.
As the island is also called "Ireland" the term Republic of Ireland is used to differentiate between Ireland the island and Ireland the country, and between Ireland and "Northern Ireland". Unfortunately for that name, Donegal- in the Republic- has a chunk of land that includes Malin Head, which lies further north than any part of NI. In fact most of Donegal is further north than Belfast. So it's kinda silly saying Southern Ireland goes further north than Northern Ireland. If that wasn't confusing enough, Northern Ireland is part of the province of Ulster but is not actually Ulster. Ulster has 9 counties, 3 of which are in the Republic and 6 of which are in NI. Remember that when you hear "Ulster says NO!" (Well 3 don't...
Do they ever say yes?)
Lastly the term Eire went out of use in 1948 when we became a Republic and left the commonwealth officially (had done so practically in 1921) and is generally taken as a derogatory term when uttered as the name of the country by British people (as in we refuse to recognise your Republic as such a State). Most people are ok with such misquoting but it does grate a bit... if you want to endear yourself to the locals bear it in mind.
Éireann is acceptable only if you are a Gaeilgoir speaking as Gaeilge...
Ireland, Southern Ireland, Northern Ireland, Ulster, Free State, Republic, Eire, Oireland, Éireann...
My head hurts....
The only really important thing to remember is this Island is made of two distinct parts. The part that is Cork, and the part that is Not Cork.
Peace and love.. (except to the fella who thinks we are a Mad Max movie based in Afghanistan...)