The term African-American was advanced in the 1980s to give Americans of African descent an equivalent of German-American, Italian-American, and so on. The term peaked in popularity during the 1990s and 2000s, but today it is often perceived as carrying a self-conscious political correctness that is unnecessary in informal contexts. In informal speech and writing, black is often preferred and is rarely considered offensive. Colored, an old term for African American people, is now considered offensive, and negro has fallen out of favor among younger black Americans.
When using the term African-American as a phrasal adjective preceding the noun it modifies (e.g., an African-American woman), be sure to include a hyphen. When the phrase functions as a noun or an adjective phrase following what it modifies, no hyphen is needed.
As of 2011, black is not an offensive term for Americans of African descent. For example, these major publications don’t hesitate to use black instead of the more politically correct African-American:
By contrast, St. Louis County, which rings the city, noted an increase in its black population of 39,000. [New York Times]
Black leaders say those in the black community understand the jobs program will benefit them. [Boston Globe]
Today Washington has a large black middle class, but when I first moved to the city in the summer of 1961, it was something else altogether. [Washington Post]
African-American has a slightly loftier tone, but it’s sometimes appropriate—for example:
The exhibition looks at how shared racial persecution shaped relationships between the Jewish instructors and African-American students. [Chicago Tribune]
As historic as Barack Obama’s claim on America’s most famous address might be, he is of course not the first African American to work at 1600 Pennsylvania Avenue. [Sydney Morning Herald]
The novel is based on relationships between white families and their African-American maids in the segregated South of the 1960s. [USA Today]
Note the hyphenation of the phrasal adjective African-American in the first and third examples, and the nonhyphenation of the noun phrase African American in the second example.