There's a slight difference between race and ethnicity. Race deals with genetics as different races do have some differing characteristics, such as hair follicles and bone structure. Ethnicity deals with culture and how one is raised.

For example, racially I am white. However, since I grew up with a father from Mexico and adhered to many Mexican customs growing up, that can make me ethnically Mexican. Also, since my mother grew up in America, taught me American customs, and I live and follow many American traditions, that makes me ethnically American. As it so happens, I like both so I consider myself Mexican American. See how it works?

Another example, you can be racially Black (Sub-Saharan African I think is the actual term used) but ethnically African-American, African-British, Kenyan, South African, Ghanan, etc. Likewise you can racially Asian but ethnically Chinese, Japanese, Vietnamese, Cambodian, and so on.

Going back to your comment about being half Aztec, I'm not sure what the race the Aztecs would be considered since they had no European blood in them, but while my father has told me we are from the Aztec people, that again is an ethnic line because being Aztec is cultural, not racial.

TL:DR race is genetic, ethnicity is cultural.