Contacting professors at the schools in which you are interested is probably the single most unique thing about graduate school admissions. Though Andrea Average never would have considered this, it is an extraordinarily valuable experience. Many positive things will emerge from your efforts, including your own evolving recognition of the application process as essentially interpersonal, not impersonal as it most often is viewed. 

As a practical matter, you probably should hope to contact at least one professor at each school to which you are thinking of applying. With seven to ten schools, this is still quite a lot of letters, but after the first is completed the rest will flow relatively easily. 

First, do your homework. Check through the brochures of all the schools you are seriously considering. Using the faculty listings and research interests as a guide, check whom might you be especially interested in working with, or under. Even if this professor does not end up being your advisor, you will have engaged a important collegial relationship, and gained useful information as well. Certainly it won't kill you to be wrong about whom to choose, but you might as well be right. 

Next, go to the largest library available. A large regional library will do, but a strong university system is even better. You are going to be doing a small bit of research on each of these professors for whom you haven't got sufficient information. Check especially through such things as The National Faculty Directory and the Directory of American Scholars. You don't need a great deal of information, but seek useful tidbits. For instance, working with a professor who won a significant award in your field is going to help increase your reputation for having worked under her. Perhaps peruse books or articles by selected professors -- but realize that a personally interesting professor might write extremely turgid articles, so weigh this impression lightly. The primary goal is information sufficient to glimpse the professor's interests and to make an intelligent, informed presentation in your letter. 

As far as the letter itself, you will first need to explain who the heck you are. Don't begin by apologize for writing or being interested in their program; you are grateful for their time of course, but remember that this is saving both you and the institution the hassle of a possibly unnecessary application. Discuss their research interests and why you are particularly interested in their program; the more specific you can be, the better. You'll probably also want to know whether they would be available as an advisor next year, or some such thing pertinent to your decision whether to apply to this school. Send a copy of your résumé: it will readily introduce them to you without having to come right out and say how great you are, and will allow them to judge your qualifications for the program. It's best to find out now if they think you might be overmatched by the competition, and for you either to respond or to change your plans accordingly.

Then wait several weeks for a reply. These are busy folks, after all, and no one gets awards or recognition or higher pay for responding promptly to prospective graduate students. 

When you do get a response, be happy. This is a momentous opportunity to gauge your candidacy and to correct any mistaken impressions. If they loved your credentials, thank them; or if, as is likely, they were wishy-washy, you can reinforce the more positive aspects. You can then choose either to continue the correspondence, or wrap it up and let them know how much you appreciate their help (you do, don't you?). Whichever it is, you now have someone on the faculty who at very least will recognize your name. You've also got a name to mention in your personal statement, indicating both your enduring interest in the program and the maturity of your decision to apply. And you probably understand the school or department a little better. It was an effort well spent.