Read the application essay question thoroughly. Some applications will have a very general, "tell us about yourself"-type question, while others will have something more specific and unusual, such as "If you had to choose one item to bring to our school, what would it be?" Different questions will require different focuses for your brainstorming, and possibly different organizational techniques.
Brainstorm some ideas about the essay question and how to connect it to your qualifications and accomplishments. Think about the different ways you can organize your essay based on the question, such as using a narrative to illustrate your accomplishments or describing each of them in a paragraph.
Write a first draft. Don't worry about editing; just write as things come to your head. Make sure it's longer than you'll need; it's much easier to cut material than it is to add it and still keep a good flow.
Take a break and let your head clear, then proofread your draft. This will allow you to catch mistakes and bad writing that you might have missed if you'd edited immediately. Make sure your essay has a clear point and flows from beginning to middle to end. Cut out any material that wanders off-topic.
Have a friend read your essay. Ask him to pay special attention for spelling and grammar errors, since it's a lot easier for someone else to catch them than for you to see them in your own work. Proofread several times---and get more friends or family members to help if you think you need it---in order to avoid the little mistakes that leave bad impressions.