How do Jem and Scout change during the course of the novel? How do they remain the same?

Jem becomes more mature.  He is still a boy at the end of the book, but he is working his way into manhood.  This is shown through his reaction at the trial of Tom Robinson (he actually cries because of the injustice of the court) and also in the way he doesn't want to play games with his younger sister (who he still sees as a child).  Scout is still a child at the end of the novel, but she is being to see things from a different perspective.  She begins to realize that things aren't always what they appear to be (Boo Radley and Mrs. Dubose are both proof of that).  The above examples could be used in an essay, but if you're talking about the actual structure of an essay I would suggest you set it up like this:

Your introduction would mention your thesis (which would be to prove that Jem and Scout change/remain the same throughout the novel) and then very briefly discuss your three main examples that prove your thesis.  Your body paragraph would discuss these examples in further detail.  For example, you may want to compare and contrast Jem and Scout's reaction to Mrs. Dubose and the lesson that both of them learned from her.  Your conclusion should reiterate what your essay discussed and possibly (depending upon what your teacher is looking for) give your own personal opinion.  The structure of your essay also depends upon the assignment that you were given, but I hope this helps.