Avoid stating the obvious. It may be tempting to begin your essay by saying you love studying history (or math or economics) and you’re applying to a program because you think it will be a great place to learn. Don’t. The very fact that you’re applying already tells the admissions committee all of that, and just about every other applicant is applying for the same reasons. Make the opening of your essay stand out so you can stand out from the crowd.
Tell a story. A good way to make your essay memorable is to start with a story. Think of an anecdote, which explains why you have a passion and talent for your field and write about it briefly in your opening. Done well, this can make your application stand out among those of a large pool of applicants, and shows the committee you are genuinely passionate about your work. Dig deep to find something valuable -- it's worth it.
Respect any page or word limits required by the application. Not only are admissions committees too busy to indulge extra verbiage, they are also looking for potential team players among the applicants. Show that you know how to follow directions because it can only help you.
Aim your essay at your audience. Remember who you’re writing for: a committee of faculty members at the schools to which you are applying. They want to see if applicants have the drive, maturity and preparation to succeed in graduate school and they also want to choose applicants who can work well with others in the department. Have your essay -- particularly the opening -- demonstrate that you are both collegial and intelligent.
Be yourself. Faculty members on graduate admissions committees are people, too, and they want to understand their applicants as flesh-and-blood individuals. After all, they may end up working closely with some of them in the future. So while you want to be on your best rhetorical behavior in your essay, don’t be stuffy. Write in straightforward, nontechnical language, not academic jargon. Your essay is your one opportunity to let your personality shine in your application -- don’t waste it.