In Topeka, Kansas during the 1950's schools were segregated by race, the Brown family believed segregating schools by race violated the 14th amendment. So they took their case to court, the Federal district court thought that segregation in public education was harmful to black children, but because segregated schools had similar buildings, transportation, curricula, and teachers, the segregation was legal.
Then the Browns took their case to the supreme court saying that even if the facilities were similar, segregated schools would never be equal to each other. This made the Court decided that state laws requiring separate but equal schools violated the Equal Protection Clause of the Fourteenth Amendment.
This case affected The united states by stopping segregation in schools and made things more equal.
Sources: http://landmarkcases.org/en/landmark/cases/brown_v_board_of_education and http://billofrightsinstitute.org/educate/educator-resources/lessons-plans/landmark-supreme-court-cases-elessons/brown-v-board-of-education-1954/