Where do I check to see my status if I am an American citizen or a resident alien in the USA?

You mean you don't know? There is a third category you didn't mention. Illegal alien. If you were a legal alien you would have papers. Were your parents American Citizens at the time of your birth? Were they EVER American citizens? Where was you born? Lots of important questions need to be answered before anyone could guess.

If you don't know you have to assume you aren't until you can prove that you are.

What does your birth certificate show as being your place of birth? Do you know if you were born in the United States?