Choose the prompt you are most comfortable addressing. Many applicants base their prompt decisions on what they think reviewers will want to read, or what they think is a more "professional" prompt. Ultimately, the prompt to which you respond is not nearly as important as your ability to respond to it honestly and expertly.
Research the institution for which you are writing the essay. This will provide you with a basic understanding of the principles that guide that institution. Whenever possible, your essay should mention or at least reflect these principles as a way to demonstrate your connectedness to the institution.
Outline your response. According to writing expert Paul V. Anderson, author of "Technical Communication," most readers of application essays search for some concept of closure and tidiness. This means employing the "bookend" essay format, in which you open with a quote, story or idea that will introduce the theme of your essay. Upon closing the essay, you will revisit the quote, story or idea so as to "bookend" your essay with it.
Draft your essay. Though not thesis-driven, your essay should have a recognizable main point, whether it be your future goals, how perfectly you would fit in at a particular school or why your teaching style is best described the the color orange, all of which are cited as typical application questions. Upon introducing your main point, offer two to three supporting points, each with its own dedicated paragraph.
Edit your essay. Maintain an active voice throughout your writing.
Polish your essay. When possible, indicate how your experience makes you an ideal applicant, and your future goals will make you an ideal member, employee or student of the institution to which you are applying.