In To Kill a Mockingbird what is the purpose of the scene between Cal and Lula at First Purchase?

This scene serves a couple very useful purposes in showing the
cultural and societal conflicts that existed in the South during the time that the novel is set. 
Even though it is a very simple scene--Lula confronts Cal about bringing white kids to church,
and is quietly put in her place--it actually is layered with deep meaning and
significance.

First of all, it shows the attitude that some black
people had at that time, which was one of isolation and anger at the white culture that was
repressing and discriminating against them at the time.  Lula felt that the black people should
keep themselves separate and not condescend to work for white people as lowly servants. She had
too much pride for that, and a more segregationist attitude, she felt, would serve her people
well.  Don't mix with the whites in any way, shape or
form.

Secondly, it reflects the attitude that the white people that
employed many of the attendees of the First Purchase church did not belong in their private
lives.  The church was one place where they could be themselves and not worry about being fired
or judged by white people.  It was a place to relax, to be at ease and comfortable in their own
skins.  If white people were in attendance, then they would not be able to do that as much.  It
shows how they often were stiffer, more careful versions of themselves around white people, and
how it was nice to take a break from that sometimes.

Another layer
in this episode is that is relays important information about Calpurnia to Scout.  Scout is
shocked that Cal would speak and act differently than at their house, and at the realization that
Cal actually has a life outside the one she has seen her in.  This is an important realization
for Scout in her coming-of-age story.  She needs to realize that the lives that black people
truly led were in actuality very different than she
realized.

Lastly, this story goes to show how much Cal loved Scout
and Jem.  She stands up for them and is willing to risk the ire of some of her people by bringing
them to the church.  Also, it shows how most of the people she lived with were kind and
accepting, no matter the skin color.  Lula was an extreme fringe, whereas most of the people
there were unbiased and loving towards all people, white or black.

I
hope that helps; good luck!