Consider opening up your essay with a personal anecdote that is related to the skills necessary for the degree you are working toward. If you are applying to medical school, for example, include a story about how you studied biology with a research scientist for an entire summer. Make sure the story truly exemplifies the skills that you are stating it does and is not just an over-exaggeration of a commonplace event.
The school will want to know why you want to attend graduate school. Some people go to graduate school to defer paying their loans or because the job market is bad. The school does not want to know about that. Focus on how going to graduate school will not only enrich your knowledge of and passion for your chosen subject area, but discuss the specific programs the institution has that you will participate in to develop your skills.
Another option is to focus on the main reasons why you should be admitted into graduate school. To say that you are motivated, dedicated and successful is not enough. Instead, tell the admissions staff that not only have you achieved high grades in the classroom, but you have also excelled in various extracurricular activities that are relevant to your career goals. Explain why you feel that this institution is the best one to further your attributes and how you can contribute your skills to the school.
Talk about where you see yourself in five or 10 years. Explain why you are unable to get to that position now. Perhaps you really love literature and want to teach the subject, but you feel that you need a deeper understanding of certain authors before you can do so. Tell the admissions staff why their institution will help to bridge the gap between your present and the future you want to attain.