In the USA the answer to this question changes every week, so follow the latest news for the most up to date answer.

During the 1960s and 1970s, the U.S. had a period of replacing racist terms with positive terms. Nigger => African-American, Mick => Irish-American, Jap => Japanese-American. These terms have mostly faded away over time. Black is the modern term and is usually used as an adjective. President Obama was the first black president. Of course, just as you never say "Hey you, white man!" you would never say "Hey you, black man!" As Aisling said, you would use their name and title: "President Obama," "Mayor Jones," "Mr Smith," and "Joe Smith."

Well you wouldn't "address" them as "black" - you would use their name or title. But it's generally ok to refer to them as black- although some people prefer "African American" or "people of colour" I'm Irish and have spent lots of time in both the UK and the USA- I've never heard anyone use the term "Afro-American"

Afro-Americans

It is not as offensive as in Russian, but unless you are black yourself, do not use word "black".