To what degree has African-American culture influenced mainstream American culture, specifically in the realm of American music?
The primary influence of African American culture in music was the rise of Jazz music as an art form. Jazz took root in slave musical traditions but developed in the mid to latter part of the 1900s after deviating from ragtime and Dixieland. It began apart from the Harlem Renaissance as it had its roots in the South, most notably in New Orleans, and is probably the greatest musical accomplishment of the 1920s. Benny Goodman's Carnegie Hall concerts (debut 1938) with mixed-race bands that included, for one, Lionel Hampton, made jazz a legitimate entertainment and music style (The Columbia Electronic Encyclopedia, 6th ed. 2007, Columbia University Press). Rock and hip-hop, mentioned above, are both off-shoots of the Jazz revolution. Jazz was and is uniquely American and, moreover, uniquely African American.
Jazz was originally by its very nature, uniquely black. At a time when segregation was practiced throughout America, Jazz was performed by black artists for white audiences. Among the famous performers were Louis Armstrong, "Big Mama" Thornton, Jelly Roll Morton, Frankie "Half Pint" Jackson, and Duke Ellington. The phrase "mojo" often heard in modern jazz music, is African in origin. Elvis Presley became famous not because of his abdominal gyrations as many perceive, but because he was the first white American who could master the style in a rock-n-roll (rhythm and blues) form rather than the earlier swing form.
American music has been affected by African American music to such a degree that it would be hard to think of what American music would be like without this influence. Most forms of American music today come directly out of African American musical forms and even those that do not have been heavily influenced by those forms.
Two of the main musical genres today are rock and hip-hop. Both of these come directly from African American styles of music. Rock has been around longer and has evolved away from its roots to some extent, but it had its origins in the "race music" of the 1950s. Hip-hop is much more recent and comes directly from African American styles of music of the late '70s and early '80s.
A third major genre is country music. This does not come directly from African American music. However, this music originated in the South where it was impacted heavily by African American music. In addition, the banjo, a quintessentially country instrument, was created by slaves in the Old South.
In these ways, essentially all American music has been influenced by African American music.