A short little explanation
The Irish language is sometimes referred to as “Gaeilge” (pronounced Gwal-gah), but it is not Gaelic; Gaelige is the name of the Irish language in Irish.
Like its Gaelic cousin, both are Indo-European languages, but Irish is actually a language unto its own. The term “Gaelic”, as a language, applies only to the language of Scotland. If you’re not in Ireland, it is permissible to refer to the language as Irish Gaelic to differentiate it from Scottish Gaelic, but when you’re in the Emerald Isle, simply refer to the language as either Irish or its native name, Gaeilge.