Does being black mean that you are African?

No, being black means that you have ancestral roots that come out of Africa. Today, a black skin color can be found in several different continents, and one might associate themselves more with Jamaican or Haitian ancestry, though technically all people with a dark skin tone had family that came out of Africa at some point in history. Unfortunately, almost all of the time it was due to being enslaved.

But you couldn't call someone born in the United States an African. They are African-American. Calling someone "African" gives the connotation that they were born in Africa.