African American is a generally accepted term for United States citizens with black African ancestry. It is of relatively recent coinage, but gained support in the black community rapidly and is now in widespread usage. The term Black American remains in usage, and while most African Americans do not strongly object to its use in common parlance (President Barack Obama, for example, frequently uses both terms) you should use the term "African American" in formal speech and writing, and in informal speech if you are not sure of the sensitivities of your listeners.
Origin of the term
Terms for Blacks were given to them by members of the dominant culture and were not the way the African descendants referred to themselves, nor did they follow same pattern as for the naming of other groups of Americans. In light of this, in the early 1980s black activists proposed a name that followed the same pattern as that of other "hyphenated Americans". High profile people such as Jesse Jackson are credited with helping to spread the use of the term. It is interesting that "African American" caught on so quickly, because by the closing decades of the 20th Century, the term Black American, a legacy from the Black Pride Movement, was the undisputed term of choice.
Who is an African American?
Most, if not all, traditional words for describing Black Americans have generated some controversy, and this is no exception. Defining who, exactly, can call themselves or can be called an "African American" can also raise some eyebrows or indeed, outright argument.
Historical definitions of "black" persons
Generally, "African Americans" are persons with visible black African heritage who were born in the United States. However, because of the history of blacks in America, the social construct of race and the longstanding division of human beings into separate races for various purposes, identifying someone as a member of the group of American blacks was often not merely descriptive, but pejorative. Categorizing black Americans inevitably led to the uncomfortable notion of eugenics. Definitions of "African Americans" have ranged from describing any person with any known black ancestor at all, no matter how far back in their lineage, as being "entirely black", although the fallacy of eugenics meant that this definition was not used for accomplished persons with any known black lineage. Another method of determining an American's race was to select a fraction of the person's blood line, traditionally one-wight or one-sixteenth; any concentration greater than that made a person "black", and therefore second-class; if any less, they were "white", with all the rights and privileges pertaining thereto.
The term "African Americans" and other ethnic groups
The situation of immigrants, Africans and other blacks who move to America and take up residence, is a challenge to this term. While the matter is sometimes hotly debated, it seems generally accepted that "African Americans" are persons born in America. Immigrant blacks are usually referred to as "hyphenated Americans" from their native land: Nigerian-Americans, Jamaican-Americans, and so on. But this does not completely solve the problem any more than it does for Caucasians: are both sixth-generational Americans of Irish descent and the person who has just landed on the U.S. shore "Irish-Americans"?
Another controversial usage is that of African people who are ethnically Caucasian or some other ethnicity. The question of how best to describe white South Africans in the U.S. comes to mind, and there is no easy answer.
Yet another consideration is how to categorize persons with mixed ethnic heritage. While there is now increasing use of terms such as "mixed-race", "mixed ethnicity" and "biracial" in an attempt to be inclusive and more accurate, these terms do not entirely settle the matter. For example, most Americans, on merely hearing President Barack Obama's ethnic makeup (White American and Kenyan), would likely describe him as "mixed-race". On seeing the President, however, a man of medium-brown skin tone, he is more likely to be identified as African-American; Obama refers to himself as such. Other people's choices are not as easy, particularly when their complexion is light.
So, describing and categorizing African Americans today is a ticklish business that demands sensitivity and understanding.