Actually the Irish prefer their language to be called "Irish" or "Irish Gaelic".
There's also Scottish Gaelic, which is the more commonly referred to as Gaelic, and as near I can tell, the Scots don't really care about that detail so much, but the Irish do.
Irish and Scottish Gaelic are definitely related, but they're still different... I think it's much like the various dialects of Spanish (Argentine vs Mexican vs Spain etc ad nauseum).
I specifically want to learn Irish Gaelic.
The Goidelic languages (also sometimes called, particularly in colloquial situations, the Gaelic languages or collectively Gaelic) have historically been part of a dialect continuum stretching from the south of Ireland, through the Isle of Man, to the north of Scotland. They are one of two major divisions of modern-day Insular Celtic languages (the other being the Brythonic languages). Goidelic is generally divided into: Irish (Gaeilge), Scottish Gaelic (Gàidhlig), and Manx (Gaelg). Shelta is sometimes mistakenly thought to be a Goidelic language when it is, in fact, a cant based on Irish and English, with a primarily English-based syntax.