What is the idea of the "background" Miss Maudie mentions on page 236?

The quote you're referring to is from Miss Maudie:  "'The handful of people in this town with background, that's who they are'" (236). 

This is in response to Aunt Alexandra's question about how the town could let Atticus stand up to intolerance and racism when it tears him up inside so much.

Miss Maudie uses the term background to refer to those tolerant and unbiased people (Judge Taylor, Heck Tate, Mr. Raymond, Link Dees, and so on) who see the racism in Maycomb and refuse to accept it, even though the majority of the town seems willing to just let things be.  In this way, those with background are paying Atticus "'the highest tribute'" because they "'trust him to do right'" (236).

Of course, the idea of "background" plays a major role throughout the novel.  Remember, previously Aunt Alexandra was explaining to Scout that she couldn't be friends with the Cunningham boy because of her background, as a Finch Scout is expected to behave a certain way.  Whereas the Cunningham boy is viewed as white trash because of his background.  Now it seems that Alexandra's view of a person's "background" may have changed some.