Along with transcripts and recommendation letters, most graduate program applications ask for some form of a personal statement. At his blog The Spandrel Shop, Prof-like Substance advises against filling that statement with clichés, saying "don't relate your passion for science by including quotes from Nietzsche or Mother Goose" and "don't use personal experience as a motivation to stamp out disease — we get it, lots of people have diseases." Above all, he adds, "don't write about your child science inspiration."
As for "do"s, Prof-like says a grad school personal statement "should be clear and concise: Get across that you have research experience, you have learned something about how science works, and that you want in on the deal." When applying to a particular lab, he adds, "talk about the work being done in that lab and where you would be interested in contributing."