What's all this "African-American" in all the primaries, American media, just everywhere American.
It's such a clumsy,multisyllabic term. What's wrong with 'black'? Nobody in public life there is using it: everything and everyone is 'African-American'.(Not quite 'everyone': Barack Obama was described originally as not African-American enough for the African-Americans, but that's political analysts for you.Best not take their stuff literally!)
What happened to "Afro-American" ? It was shorter.
In Britain the correct adjective,and the one used of themselves, is 'black'.
The term Afro-Caribbean is in use. It's used by the community of black Caribbean ancestry to describe their community precisely. That apart, they call themselves 'black' and not British Caribbean,or Anglo-Caribbean, or Anglo-Trinidadian etc
We don't have Africans, Anglo-Africans (or Afro-Africans

)either. People who are of direct black African ancestry call themselves 'black' too.
And, of course, we have Asians.These people are not oriental. They are from the Indian sub-continent. (Anglo-Indian is either a style of cookery or furniture or a historical reference to people, in the days of the British Raj, who were of half-Indian and half- British parentage)