An ethnicity (not a race) that is equivalent to "Asian American" or "Latino/Hispanic American" and is used to designate the population descended from enslaved Africans in the US (and sometimes the Americas in general). It is not synonymous or interchangeable with "black" which is a racial category and may include African Americans (formerly Afro-Americans or American Negroes), Afro-Caribbean people, Afro-Latino people, or any other group that may be racially defined as black. The choice to identify as African American has nothing to do with nationality or race, but with a unique cultural heritage that developed in the US. Therefore, the critique that African Americans are not "African" because they were not born in Africa is irrelevant and as nonsensical as saying that a 10th generation Asian American is not Asian because they weren't born there- the titles are meant to recognize recent origins, current cultural practices, and identities that are distinct from mainstream white American culture (although its forms are often appropriated by mainstream white American culture).
White person who's terrified of difference and of really thinking about history because it means she has to acknowledge that her ancestors were pricks: "You're no more African than I am European."
African American: "That's true but the implications of someone looking at us and guessing that my ancestors were African and that yours were European are pretty freakin different, asshole. Read a history book and maybe you'll figure out why."