A short history lesson. It used to be offensive to call people black or nigger. Instead, the common place term was negro. Many older black people refuse to recognize their race as anything other than negro, so much so that the last US census had negro as an option.

Later, the term black seemed to switch places with negro as the appropriate term. Negro, by itself, is not particularly pejorative, but black is much more accepted. It seems discussing a black person's race, regardless of terminology, tends to offend some people.

Personally, I hate the term African-American. Unless you moved here from Africa recently, you're just American.