Cornwall, is it part of England or a country in it's own right?

I have friends in Cornwall who are extremely defensive of the fact that Cornwall is not part of England and is a country in it’s own right that is, like Wales for example, part of the United Kingdom. I have done some brief, online research (ok, I checked Wikipedia!) and can’t find anything that says Cornwall is a country. It all says that it is a county in southwest England. I know they have their own language (although I wonder how many people ACTUALLY speak it anymore) but that doesn’t actually make a country does it?

Side question – what does make a country?