Assimilation is the process in which one, usually minority, cultural group comes to be more like the majority group in terms of its cultural traits. Biculturalism is the absence of this process. In a bicultural situation, the minority group does not change its cultural traits.
With respect to African Americans, assimilation would involve African Americans coming to have cultural traits that are the same as those of white Americans. They would dress the same, listen to the same music, eat the same food, and speak with the same sorts of slang and accents as white Americans. Biculturalism, on the other hand, would be a situation in which African Americans had a very distinct culture. In such a situation, all of the cultural traits mentioned above would differ between African American and white American culture.
What we have in our country is a mixture of these extremes.