Start the essay writing process as early as possible. Even before you begin to apply, perhaps during your junior year of high school, or the summer prior to your senior year (the most appropriate time, in my opinion), you can download applications on the web and 'mull' over some typical essay questions asked by schools that interest you.
Free write...the rewrite. Those of 'us' in the writing business know that the key to great writing is 'rewriting.' Begin first by taking time to write something down; a thought, a story, a life experience, whatever comes to mind. Then begin the process of reviewing, editing and reflecting. Write multiple drafts, and elicit objective reviews from one or two people that can provide constructive feedback (teacher, counselor, or mentor). Be sure to allow time for those that read your essays to review what you have written (hence...begin early).
Admissions committees seek students that present intelligence, maturity, and awareness. You can best demonstrate these qualities by following the guidelines set forth for your essay(s). Keep the length below established word limits, and stick to crisp, classical writing structure: well organized, strong introduction, and reflective conclusion.
Avoid the temptation to embellish your essay with achievements. It is appropriate, for example, to elaborate upon a single experience or achievement and relate that 'one' event to your application for admission (experience led to personal growth etc), but it is entirely different to splash the essay with items that are listed elsewhere on your application. It is a waste of time and a waste of what I refer to as 'quality character real estate.'
If you have experienced tragedy or hardship in life (which many of us, including myself, have as young people) be very careful about how you present those personal problems and traumatic events. If you really have not 'overcome' and grown from them it can backfire. It can be done effectively, and you should never shy away from telling your story, but you have to remember to be positive. Focus on what you have learned, what you have gained, or how you have grown or changed, from difficult experiences in life. Remember that adversity is something to be overcome - so, if you choose to write about something difficult or dark, talk about how you overcame the situation and how you will continue to grow in the future.
Avoid the urge to use language that you are unfamiliar with: use the essay as a dialogue. Some of the most inspiring essays I have ever read are void of any sense of adventure or eloquence. They were 'real:' a student talking about learning the values of hard work by working a job after school to help support their family, or talking about their passion for art, children, tall buildings. Regardless of the story or theme, inspiring essays commonly present positive thinking, intriguing perspective, and personable vocabulary.
Last but certainly not least, do not stress over the value of your admissions essays. An admissions essay is not nearly as valuable as three to four years of hard work - so, don't forget that if you find yourself staring a blank Word document for some time when you attempt to write yours. Your essay will not overcome a poor grade average at many competitive institutions. With the said, do remember that the true value lies in what can be learned about you - present the person behind the academic record and test scores and your essay will serve you well.