Most admission essays, academic essays, and scholarship essays are designed to teach something to the reader. In truth, if you are writing an essay that involves class material and your teacher is the reader, your teacher may already know what you have to teach. So, you will write as though you really are saying something new. Who knows--maybe for your teacher, it really is!
Admission and scholarship essays normally instruct the reader first of all about you, whether directly or indirectly. Even when your topic is about something else, such as your favorite role model or the best way to eat spaghetti while blindfolded, you are teaching your reader about yourself: this is what I find interesting or valuable; this is how I solve problems; this is why I would be a great member of your community.
When you write to instruct, think about what is worth knowing about your subject. Then, (1) instruct the reader why this point is worth knowing, and (2) make the point.
(1) Will your reader be impressed if you compare the novel's hero to a tree? Well, it depends: are trees or forests important in the novel? Does anyone in the novel get transformed into an inanimate object? Does the character act in a "wooden" manner? In other words, if you can make a good case for the knowledge being important in its context, your reader will be interested to learn what you have to teach.
(2) You have had good teachers and bad teachers, right? There are many good ways to write an essay that instructs. You can choose to lead the reader through a chain of thoughts, provide the reader with a bunch of data that illustrates or proves a general point, explain to the reader how a particular detail fits in the big picture, compare various sets of facts so that your reader understands what's what, tell the reader a story about something worth knowing, and so on. See "Take Your Reader on a Trip" for more ideas.
