Right, time to roll the sleeves up and dive in, as an Irish person!

Here we go again - I meant to post "Hiberno-English" originally.

My interpretation of the question is that it definitely does not refer to the Irish language, which, in Irish, is called Gaeilge, as Cillian pointed out. In Ireland, we don't refer to Gaeilge in the anglicised form "Gaelic", although the Scots do, and the term is often used by Americans to refer to the Irish language.
As some of the other answerers have pointed out, the Irish language is totally distinct from the English language, in that it is a celtic language, not a germanic one. If you want an example of how different it is, just look at my first name, Aoife, pronouned "Eefa"! My surname is Ni Chinneide, which, translated, means "the daughter of Kennedy". I use the anglicised version, Kennedy, for my proz.com profile.
Irish and English are the official languages of Ireland, and the Irish language is still mandatory throughout primary and secondary school.
There are some areas of Ireland, called "Gaeltacht areas", where the Irish language predominates over English.
With regard to what Wikipedia calls "interference", I would prefer to call it "enrichment", in that some lovely expressions from the Irish language have found their way into everyday English, as spoken by Irish people - I think Hiberno-Irish is the least confusing term for this.
One example of this influence is the very common English expression "smashing!".
I read somewhere that it's derived from the Irish expression "Is maith é sin", which means "that's good", or "that's great".

PHEW! I feel like I've just written an exam.
I hope this helps.
Good luck!

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Note added at 2 hrs 26 mins (2005-11-04 01:44:55 GMT)
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I've just remembered another excellent example of "enrichment", which I only discovered means nothing to people from the UK when I used it a few months back.
When we have something to do, but we keep on putting it off/delaying it, we say "I keep putting it on the long finger".
When I said this to a friend of mine, there was a very blank expression on his face, and I realised that it must come from the Irish expression "Chuir ar an méar fada é" - literally "put it on the long finger".


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Note added at 2 hrs 45 mins (2005-11-04 02:03:52 GMT)
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And the most famous of all... "May the road rise to meet you"... a literal translation from the Irish "Go n’eirí an bóthar leat".