It's also been called the black aesthetics movement and the black arts movement.  Basically, though, the black cultural movement, as I'll call it here, was at its height in the latter half of the 1960's and on into the 1970s, though echoes of it are definitely still around today: just listen to Lauryn Hill's music, watch Spike Lee's movies, read Toni Morrison's novels, or see August Wilson's plays.  If the movement has a single central purpose, it's to make African-Americans (and, indeed, black people the world over, if it can) totally and irreversibly proud of their racial and cultural heritage.  One of the most commonly recited mantras of African-American activists in the late '60s and 1970s was "black is beautiful," an idea unexplorable to black Americans of previous decades. 

There are a few other beliefs central to the black cultural movement.  The first is that art and politics are inseparable -- that great art is born from political strife and can actually be an important tool for raising consciousness about the social problems oppressed people deal with.  It's not that this is an especially original idea in the late 1960's: writers like Frank Norris and Upton Sinclair had held way back at the start of the twentieth century that art could help bring about a socialist workers' revolution, and their own novels (e.g. The Octopus and The Jungle, respectively) had tried, however unsuccessfully, to make that revolution happen.  In the 1960's, though, black writers and artists bring this idea of the marriage of art and political change roaring back again, arguably to greater effect than the socialists ever had.

A second idea big in the black cultural movement pertains to "black nationalism," as it came to be called -- a doctrine holding that if, in fact, black people have only generally suffered by their exposure to white society, they should form a separate society having as little as possible to do with whites.  This belief, powerfully expressed in the politics of the Black Panthers and the sermons of Malcolm X (the earlier Malcolm X, at least), was deeply contrary to the "integrationist" agenda of many American Civil Rights supporters (Dr. Martin Luther King included) and was descended from the ideas of Marcus Garvey, an African-American intellectual who suggested in the 1920s that black Americans move to Africa to found a new black-only nation.  In any event, much art by black Americans in the 1960's deals in some way with this notion that separatism is preferable to integration, and that black people will be better off if they avoid the dominant white culture and rid their minds of the imperialist myths and standards white society has filled them with.

A third key belief is that there is something fundamentally, irreducibly black about black people's art and culture, even if that "something" is hard to name or pin down.  If European-descended culture often seems prim and repressed, African-descended culture will be more sensual ("funky," Toni Morrison would say).  If European-derived art is shaped by restrictions of form and propriety, made into definitive, finalized "texts," then African-derived art is looser, freer, more improvisational, and closer to an oral tradition that never totally fixes or finalizes artworks.  (You could think, for instance, of the difference between "classical" music and jazz.)  If this is true -- if there is something different about black perception, black art, black culture, black language -- then only black forms of music, language, dance, painting, etc., are capable of expressing black experience.  It's only fitting, then, many black thinkers from this movement said, that black artists use black language and forms in their works, no matter how improper or unacceptable a white art establishment might hold them to be.