General: What part of the cow does corned beef come from?
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.
There are rumors that corned beef comes from -- let's put this delicately -- the "lady parts" of a cow. These are untrue. Corned beef is a muscle meat, not an organ meat, and anybody who tells you otherwise is just trying to gross you out. .
Since we're a website about European foods and cuisines, there are two questions we get a lot:
(1) Is corned beef Irish? Yes. People here started making it around 800 AD, and even wrote poems about it.
(2) 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" (it's pork, really) and cabbage. They were having trouble getting the brined pork that the dish is based on, and started using salt(ed) beef as a substitute. (There are claims that their Jewish neighbors in the tenements of lower Manhattan put the Irish immigrants on to the beef, and though this is entirely possible, there's no proof.)
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. We particularly recommend Irish Coffee cake and the Baileys marble cheesecake.
You might also like Peter's (Irish) Mum's soda bread recipe, which frankly is the best on the Web. (EuroCuisineLady says: 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.)