First of all - forget just about everything you know about essay writing. Of course there's a 'right' way to do it - the way that will get you a good grade, the way that's expected in class even through grad school.
Clear Thesis Statement, Clear Outline, Each Chapter has one Topic Supporting the Main Claim and Introduced by a Topic Sentence, etc.
In school, you also need to have good ideas, be logical and organized... and stay within 'acceptable' thought. In my experience as a judge for essay writing contests, they winners usually break those rules. So if you want to win big money entering essay contests, here are a few steps:
1) Be OUTSIDE the box. Charm, surprise, shock. Everybody else is going to be writing about the topic - "Global Warming" for example. Saying boring (yet true) stuff about how global warming is bad. You need to stand out from all this boring stuff by taking a novel and unprecedented approach. Talk about something specific, funny, that has little to do with global warming.
2) Make it PERSONAL. Make it about you, yourself - not 'some kids out there somewhere'. Talk about your real life and experiences, things you've actually dealt with.
3) Direct it at the reader. "You" is the most powerful world. Talk about 'I' and "You", as if you're having a conversation with the reader. "You don't want the world to end, do you? Of course not."
4) Be funny, and informal. Formal = boring. You can (and should) be informal, use common speech and slang.
5) Write Well. Good writing is about
painting a picture. Use a lot of very specific examples and set the scene; uses descriptive words and adjectives, don't talk about concepts, talk about your bathtub, the green scales of a lizard, or a single drop of sweat; people can see and feel images in a way that they can't feel words.
Above all, have CONFIDENCE and have FUN. Enjoy what you're writing - don't be worried about what the judges think or whether or not you'll win. Like Hitch says -- the objective is to shock and awe.
