The two styles that originate in Ireland are Irish Dry Stout and Irish Red Ale. The more famous of the two is, naturally, the Stout, since that’s the one everyone is exposed to.
The Dry Stout style evolved from the London Porters of the time, and were a stronger version of that style—a "Stout Porter." It was also a stronger beer than is (typically) brewed these days—around 7.5% alcohol by volume (compare to 4-6% currently).
I won’t go quoting numbers excessively, but I’ll point to the BCJP guidelines as a starting point.
The Irish Red Ale is a lighter beer, a maltier and less hoppy counterpart to the English Bitter style. Easy drinking, with toasty notes and a possible dry finish, Michael Jackson in Ultimate Beer pairs Irish Reds with pork when considering food:
What could be heartier than an Irish red ale (even if the color is closer to chestnut or amber) with boiled bacon and cabbage, or with loin of pork, perhaps? The love of bacon, ham, and roasts is not merely a question of Irish tradition: there is also in Irish ales a sweetness, a creaminess, and sometimes a slight butteriness, which highlights the flavor of such dishes.
Again for the numbers junkies, the BCJP guidelines for Irish Red; and looking at the BeerAdvocate top 10 list for it, it’s again dominated by American brews (though overall less reviews than Dry Stout), with one exception: The Biddy Early Brewery in Ireland.
(Yes, there could be a whole discussion on the preponderance of American brews on these lists for foreign styles, but that’ll be a discussion for another time.)
Irish Red is definitely a less common style than Dry Stout; other than seeing (off the top of my head) Killian’s Red (which isn’t technically a true Red Ale) and Smithwick’s at the store, they seem to be virtually non-existent from the shelves. Although in a number of ways the style sounds very similar to the American Amber style of ale, so a reasonable substitute in that case is relatively easy.