The republic of Ireland is a description of the country, it is not the official name. The country is called Ireland and it is a republic. If you look at the entire rest of the site you linked to Tee, they refer to the country as Ireland.
Eire is written on the stamps because they are in Irish. If you are speaking or writing in Irish then Eire is the correct term. But not in English.
Southern Ireland is patently ridiculous since a) its not the name of the country, b) lots of the Republic is in the northern half of the country and c)calling 5 sixths of an island southern, even the northern bits is just silly. Oh and d) Southern Ireland means Munster.
The reason its important is that the name of the country has changed many times, and the names have political and cultural significance.
Since people seem to have read the Wiki page without actually reading the part about the name, I'll reprint it here;
Article 4 of the Constitution of Ireland, which was adopted in 1937, provides that:
The name of the State is Éire, or, in the English language, Ireland.
For all official purposes, including international treaties and in other legal documents, where the language of the documents is English, the name of the country is Ireland. The same is true in respect of the name Éire for documents written in Irish.
The Republic of Ireland Act 1948 provided a supplementary description of the state as "the Republic of Ireland" (Irish: Poblacht na hÉireann).[8] The Act was to change Ireland to a republic rather than a form of constitutional monarchy and transferred authority from the monarch to the elected president. No change of name took place due to that act, and in 1989 the Irish Supreme Court rejected an extradition warrant that used the name "Republic of Ireland". Justice Walsh ruled: "if the courts of other countries seeking the assistance of this country are unwilling to give this State its constitutionally correct and internationally recognised name, then in my view, the warrants should be returned to such countries until they have been rectified."
If you are speaking english, the name is Ireland. If you are speaking Irish, the name is EÃre. If you are describing the country, you can use the Republic of Ireland. In practice you can say whatever the hell you like, as long as you know some people will not like it, and its really quite rude.