Understanding the setting is a key to understanding racism in To Kill a Mockingbird. Keep in mind that this fictional book is set in an historically accurate small southern town, post-slavery, but pre-Civil Rights. While black people and white people have been constitutionally granted an attempt at equality, the reality of the situation is that they are still very much unequal. This book takes place during the time of segregation and Jim Crow Laws. As a result, there is mostly racism of white people toward black people, but there is also racism of black people toward white people.
One predominant example of racism of white people toward black people is the outrage of the town toward Atticus for defending Tom Robinson, simply because he is black. Along with this attitude comes the use of the term "nigger" whether as a means of a direct insult, or simply as a common yet degrading term:
Then why did Cecil say you defended niggers? He made it sound like you were runnin' a still. (75)
"Atticus," I said one evening, "What exactly is a nigger-lover?" (108)
Both Jem and Scout hear the term "nigger" from the kids at school, their cousin Francis, and their old neighbor, Mrs. Dubose. They do not even know what it means, but they can tell from the tone that it is an insult both to black people and to their father.
In addition to racism of white people toward black people, there are also examples of racism of black people toward white people. When Calpurnia brings the Finch children to her black church, she is met with some opposition:
Lula stopped, but she said, 'You ain't got no business bringin' white chillun here--they got their church, we got our'n'. (119)
Lula is a black woman who is opposed to having white children in her black church and displays this in a hostile way. She is a direct example of how segregation created racism in both white people and black people.
The reference links provided will also lead you to more examples of the theme of racism in the novel.