The New York Times has an article detailing the struggle over the term "African-American." Long story short, some black people in the US do not want recent immigrants from Africa, the Caribbean, etc., to describe themselves as "African-American." Why not? Well, because the two groups arguably do not share a common heritage, and so maybe they do not deserve to be lumped together in the same catch-all ethnic category. There are also differences between recent immigrants and native American blacks in education, socio-economic mobility, etc., and these differences have political ramifications, of course.
But as Instapundit points out, if an American citizen who emigrated from Nigeria can not be described as an African-American, then what is she supposed to call herself?
Leaving aside this discussion for a moment, I see two other problems with the term African-American. First, as Baldilocks has noted, the term is frequently used to refer to non-Americans. I believe this type of use is often unconscious. Last night my roommate's girlfriend (a third-generation American of Japanese descent) called the members of a Jamaican women's relay team African-Americans. We usually don't call Jamaicans Americans, so why would black Jamaicans be called African-Americans?
Another example: when I was an undergrad at BYU, I touched off an angry debate in an English class by correcting someone who spoke of Othello as an African-American. (I think this term is especially inappropriate for him since he was a Moor, and thus perhaps not of sub-Saharan African descent.)
So the first problem is the (common) use of the term to describe those (non-US citizens) who are by definition, excluded. A second problem, pointed out by a white South African during the discussion in my English class, is that the term includes, by definition, people like himself. And nobody means Boers when they say African-American, even Boers who have become naturalized US citizens.
So African-American has definitional problems. And for years I thought that this was why I preferred the term black. But black has very similar problems, definitionally. Colin Powell is "Black," even though he is, at best, tan. Sri Lankans are not "Black," even the ones who are quite black.
I guess I just like the sound of black better. Maybe it's because I can't imagine James Brown singing, "Say it loud / I'm African-American and I'm proud." (Although if anyone could make that sound good, it would be the Hardest Working Man in the Show Business.) Or maybe it's because I agree with Orwell: "Never us a long word where a short one will do."
Then again, I much prefer "Mormon" to "LDS," even though the former started out as a slur and the latter (heh) is just an abbreviation of the church's official name. Since there have been plenty of people who have asked me which term prefer, so as not to offend, the least I can do is to politely inquire and respect the wishes of my black or African-American brothers and sisters.