I think Lula's character helps illustrate how much of a protective mother figure Cal is to the children. Notice how she fiercely protects the children when Lula questions her as to why she is bringing them to their church. This also foreshadows the racial tension that is building up because of the looming Robinson trial.
What the scene also does is to illustrate the ways in which the black community has been so segragated from the white community. Calpurnia works amongst the Finches, and Jem and Scout have always accepted her as a part of their world. But there are no other African-Americans that exist in their world. This is because they have been pushed aside, out of sight if you will. Lula is upset because the whites won't allow the blacks to be a part of their community - therefore, the whites shouldn't try to be a part of the black community. It is a matter of pride - you don't want us, so we don't want you. Of course, some of that will change with the trial, as Jem, Scout and Dill are permitted to sit with the negros in the balcony of the courthouse.
Lula is a black woman at Calpurnia's church who is indignant that Calpurnia has brought Jem and Scout to their Sunday service. She rudely says, "You ain't got no business bringing' white chillun here - they got their church, we got our'n" (Chapter 12). Lula illustrates that prejudice and stereotyping is not something that is confined to one particular group, it is pretty much found everywhere. As a member of a racial group which has been discriminated against by white society, Lula turns around and gives the same unfair treatment right back to innocent individuals, solely because of their racial identity. Lee appears to use Lula to show that, unfortunately, prejudice can and does work both ways.
Lula is the black side that is hurt and upset that they are not accepted in the white community and think that if they cannot be brought into the white community, then the whites cannot come into theirs.
Harper Lee introduces Lula in her novel, To Kill A Mockingbird, for several reasons. Firstly, Lula shows that personal, social and political history affect how people behave. Lula is bitter because the white people are allowed to come to her church but she is not allowed to go to the white's church. Her history of being a subject of racism ahs made her generally angry. When Lula becomes upset at Calpurnia for bringing Jem and Scout to the Negro church, the shows the theme of the evils of prejudice and segregation in the deep South during the 1930s. Lula is prejudiced against white people because they have been racist against her before. Lula is really the only black person in the novel who is angry about her white counter-parts. Tom, Helen, Calpurnia, Zeebo, Reverend Sykes and other black people in the novel expressed little or no resentment towards white people. This, in my opinion, is a flaw in To Kill A Mockingbird because many blacks, in reality, were angry at whites and their (the black's) situation...