Take down some notes about yourself. Highlight some of your personal interests, activities and achievements that you may want to mention later on. Note any awards, certificates and major moments in academia or in life, especially moments that may have influenced your decision to attend graduate school. According to College Board, these points should be revealing, detailed and personable. Brainstorm from this list what you may want to discuss in the essay.
Remember that some graduate schools require specific topics to be written about, so it is important to pay special attention to requests they may make.
Always research the school to which you have applied---make sure your research is thorough. Mention some features about the school so that admission officials know you are interested for a specific reason. Make sure to research the school's specific departments. For example, if you want to enroll in a non-fiction writing program, make sure that the department offers that specific concentration.
Begin writing the cover letter by introducing yourself to the team of admission advisers. Start by writing a vivid, unique statement. The statement could be something like, "It happened on my third birthday. I was hooked from my very first science project." This is a lead-in to what you were doing, who you are, and why you want to study science. This is important, since it shows you are a unique candidate that goes beyond simply saying, "I would like admission to this program." Remember to be professional while writing, letting enough personality shine through.
Expand on the opening statement. Describe why you have such a passion for the topic you want to study. Explain what you learned as an undergraduate and how that made an impact on your desire to attend graduate school. Discuss personal points about that particular academic topic. Relate this to something personally meaningful for you. Continue the essay until you've told the story about why you're interested in science (for an example) and what your academic and personal goals are. Discuss what you've researched about the school and let them know how they specifically will help you realize your dream.
Edit the cover letter for correct spelling, grammar and sentence structure. Have a professor or friend read it over for clarity.