What was the judicial system like in the South in the 1930's?

This is a pretty broad question, but since your last
question was about To Kill A Mockingbird, I will answer this with
regard to that book.

The judicial system in the South in
the 1930s was (as in the book) heavily tilted against black people.  On a formal level,
blacks were treated equally by the legal system.  The one exception to this was the fact
that blacks were not allowed to serve on juries.  The Tom Robinson trial might well have
ended differently if there had been any black jurors.

Just
as important, however, was the informal bias against blacks.  More or less everyone who
participated in the judicial system would have held racist views.  This would lead to
verdicts like the Robinson one where a black witness's story would not be believed if it
contradicted that of a white witness.