It is essential to persuade admissions officers that you are someone they should admit. You persuade them by demonstrating that you are the kind of person they want in their institution. Even if you are applying to one of the hundreds of colleges that use the Common Application, the following advice is very important: learn what the institution values and show that you either exemplify or can help the institution achieve those values.
At most schools, a primary value is being able to communicate well. You signal proficiency in written communication by writing well-crafted, meaningful, believable essays.
But your essay also instructs the reader about you, both directly and indirectly. Even when your topic is about something else, such as your favorite book or the best way to eat an ice cream cone while blindfolded, you are teaching your reader about yourself: this is what I find interesting or valuable; this is how I solve problems; this is why I would be a great member of your community. Many schools, especially those in the Ivy League and other elite schools, already expect that the essays of most admitted students will be written very well. What they are looking for is a match between your qualities and values and what they value.
In addition to the specific values of each institution, just about every academic community values a lot of the same general qualities. You do not need to signal all or most of these values in your essay(s)--other parts of your application will signal several of them--but you should take care not to seem like the opposite of any of the things that the school values. Note that it normally goes over better to signal them through your narrative rather than to state them directly; avoid statements that merely announce, "I am great; look at me." The following list provides general qualities and values you could choose to express:
1. Strong traits of academic character. These are normally expressed elsewhere in an application, but a successful essay can show more details about (a) your excitement to learn, such as when you describe a particular research interest; (b) your hard work and other kinds of dedication, such as in an experience where you kept working in the face of a setback; (c) your interest in solving problems, such as when you noticed a problem and did something about it; (d) your creativity, such as when you did something unusual; or (e) your honesty, such as when you acknowledged that you had an important lesson to learn. An essay about a particular achievement, if it does not repeat too much material elsewhere in the application, might be ok so long as you use the achievement to show something new about yourself.
2. Traits that show potential for success. Good communication skills are fundamental. Beyond that, you can demonstrate that you will make good use of the resources at an institution to achieve your goals by showing explicitly (a) what your goals are and (b) how the institution's resources will help you achieve your goals. Or you can suggest implicitly that you have (a) ambition, such as when you imagine what your life will be like at some point in the future; (b) excitement about succeeding, such as when you relate how you felt after accomplishing something; or (c) skill in solving problems, such as when you accomplished a complex task for the first time.
3. Strong social skills. Remember that the institution is an academic community. Admission officers are especially hopeful that you will contribute to the overall life of the institution--in activities, in the dining halls, in the dormitories. A successful essay can demonstrate (a) your appreciation for social interactions, such as good times spent with friends and family, either in one specific incident or over the course of a season; (b) your leadership qualities (not your leadership positions), such as when you pulled people together to solve a common problem; (c) your engagement with interesting people who differ from you in important ways (don't simply discuss "diversity" but be specific), such as on an international trip or a trip to a different part of town or even a trip to your grandparents' house; (d) your commitment to community service, such as when you volunteered in a project to help people out; or (e) your interest in government, such as your involvement in a cause or your work for a political campaign on the basis of ideas about how to improve society.
4. Personal virtues. See below on "Which Personal Qualities Count As Good?"
Furthermore, it is important to understand the specific values of the institution to which you are applying for admission. For example, some schools value applicants who are really quirky but also really smart; others look for strong ethical or religious commitments; still others look for particular academic interests; and still others will admit just about anybody with the right test scores or athletic abilities so long as the essay does not sour the application. As you decide which schools you prefer, you probably will encounter information about each school's values. Many schools include these values in admission materials and on their web sites, such as in a "Mission Statement" or in a "Letter from the President." As you learn about the curriculum, student activities, and other programs, you will find out where the school is putting its energy and what qualities the school is most proud to offer you.