Hello. My name is*****'m a licensed attorney, experienced in corporate law, and I will be happy to assist you.
Very interesting questions!
A US company is any entity that is incorporated (if a corporation) or registered (if a partnership) or domiciled (if a sole proprietorship) in the US. Incorporations and registrations are made through the states, not federal government.
As for why so many are incorporating in Ireland? The answer is one word: TAXES. Specifically, the low (12.5%) corporate tax rate. That's the primary factor, but there are others. There are other places in the world with low corporate tax rates, such as Bermuda. But many companies cannot have their regional headquarters located in a jurisdiction that--whether fairly or unfairly--has obtained a reputation for less-than-aboveboard business dealings. The label "tax haven" implies sunny beaches and shady business.
Ireland, by contrast, does not conjure up the impression that a company is being sneaky in order to avoid paying its fair share of taxes.
So as for why so many companies choose Ireland, I guess the answer is two words: Taxes and Public Image. Even if the tax rates are lower in some obscure Caribbean island, it's worth it to many companies to align themselves with Ireland, which does not have the reputation of a Banana Republic.
Please let me know if you have any further questions. I hope this helps! If so, please be sure to rate my answer, since that is the only way I can receive credit.