How to write a good university application essay in the United States?

One of the requirements for applying to most U.S. universities is writing a short essay that summarizes your purpose in applying to the university.
Together with letters of recommendation, the essay helps the admissions committee distinguish you from other. It also shows how well you can write a coherent and persuasive story - an important part of being a successful student in the United States. A well-written essay is therefore almost a key factor in the admissions committee's decision. The main purpose of the essay is to get to know you as a person, so whatever the topics, they can all be categorized into three groups:
The topics in the first group can be summarized as "Tell us about yourself."
For example: "Tell us about yourself and your plans for the future"; or "Tell us about your favorite book as a child." The themes of the second group can be subsumed under one question, "Why us?"
For example: "What is it about our university that prompted you to apply here?"; or "Why do you think our university is the best fit for you?" Group three topics determine your level of creativity and out-of-the-box thinking. For example: "A man cannot be too careful in choosing his enemies (Oscar Wilde). Tell us about the relationship between you and your eternal enemies (either real or imaginary)"; or "Why did you do it?