Desegregation
Desegregation (also called integration) is a system whereby two groups which had previously been kept apart are brought back together. The most commonly thought of example (and the one Reagan was speaking of) was the desegregation of schools after the Brown Vs. Board of Education Supreme Court case decision of 1954. Following a 1896 Supreme Court case, Plessy vs. Ferguson, schools and other facilities could be legally segregated if they were equal. During Brown vs. Board of Education, the court decided that separate facilities were inherently unequal and ordered that schools be reintegrated.
There was heavy resistance throughout the southern states, so much so that President Eisenhower sent Federal troops in to Little Rock, Arkansas when the governor at the time threatened to use his National Guard to block black students' entry into the school.
Speech Relevance
In 'Encroaching Control', Reagan mentions it in the context of Federal control of the country's schools:
The former president of the National Education Association spoke publicly on the probable need for temporary federal control of the school system in order to bring about integration in the South.
Reagan was against Federal intervention overall, so he would have been against the idea of Federal control of the schools. In the end, "temporary Federal control" never happened.
