Corned beef comes from the brisket and silverside (just under the topside) of the cow. Both of these are tough cuts requiring either long, slow cooking, or pickling in brine, or both — hence their use in corned beef. See this article for lots more background on corned beef, including the oldest historical info.
Is corned beef Irish? Yes. People here started making it around 800 AD, and even wrote poems about it.
Is corned beef and cabbage Irish? No. CB&C is an Irish-American dish invented when previous centuries’ Irish emigrants to North America were trying to develop a substitute for one of their favorite dishes from home, boiled bacon and cabbage. (They were having trouble getting the brined pork that the dish is based on.)
But corned beef and cabbage has never really caught on in Ireland, and hardly anyone born here eats it. It’s usually seen as poor people’s food — or as something to feed to the tourists who keep on demanding it, thinking it’s traditionally Irish. The above article will tell you more about that, too. (Secondary to the article: once again I checked the supermarket to see if I was possibly mistaken about this. And once again I found the usual result: three packages of corned beef, eighty packages of assorted pork and boiling bacon.) One note, though: the way Irish people usually eat corned beef these days, when they eat it at all, is in sandwiches.
If you’d like to see what Irish people really eat these days, please check this page for some background. And if your heart is set on corned beef and cabbage anyway, take a look at this page, where there’s a terrific collection of real Irish desserts. I particularly recommend Irish Coffee cake and the Baileys marble cheesecake. (OM NOM NOM NOM.)
You might also like Peter’s (Irish) Mum’s soda bread recipe, which frankly is the best on the Web. (And I’m not just saying that because she was my mother-in-law. She was an artist with the stuff, having been born here just past the turn of the last century, and she knew how it should really be made.)
To all those of you about to go Drown The Shamrock: Yes, on March 17th you’ll all be Irish. Enjoy! (But be warned: when you get over here, no matter how many Irish ancestors you have, even this one, twenty years won’t be anything like enough to make you really Irish. And don’t think an Irish passport will matter: the neighbors won’t be fooled for a second. …But you would have known the job was dangerous when you took it.)
Enjoy anyway. And please don’t dye the beer green. That’s one of the things really Irish people really don’t need to do. The green is either in your heart, or it’s not. Putting it in your liver won’t matter a bit.