A narrative essay is an essay that tells a story. You could write one with To Kill a Mockingbird in the form of a book-report style essay.
A narrative essay should focus on what happened in the book. It will describe the plot, but like you are telling a story to a friend. It is not as stale and truncated as a summary. While you want to be concise, you also want to be interesting.
As with any kind of essay, you need to begin with a thesis. In this case, the thesis is telling what the story will be about and not stating an argument. As with any essay, you will still want to stick tightly to a theme.
Think of the narrative essay as retelling the story. You can and should incorporate quotes into it, but in a storytelling way. For example, if the theme you are focusing on is the importance of caring about how people act and not what they are, you might describe this.
Although Aunt Alexandra cares about where a person comes from, Atticus cares about who the person is. When he tells his children that they are from an important family, he practically chokes on his words.
"Your aunt has asked me to try and impress upon you and Jean Louise that you are not from run-of-the-mill people, that you are the product of several generations' gentle breeding-" (ch 13)
He is so frustrated with himself at passing on this information that Scout feels he is not her father and begins to cry. He tells her to forget everything he said about the facts of life. He wants them to follow the code he taught them, not Aunt Alexandra's.
This is an example of a way to tell a story and still include important details and good quotes.