Brown v. Board of Education marked the NAACP's greatest success in its fight to

The answer is: end segregation in all public education.

The brown v. board of education case only erase racial segregation in educational institution, but it did not eliminate the racial segregation in other aspects of american society at that time.

But, the case caused a ripple effect that lead to the creation of the civil rights act in 1964 that ended racial segregation in any other aspects as well.

To get rid of segregation between blacks and whites