The Statement of Purpose requested of applicants to masters and doctoral programs in the United States is quite different to the general essay requested of applicants to undergraduate programs.  For example, Harvard's general instructions for applications to its programs are as follows:

Describe your reasons and motivations for pursuing a graduate degree in your chosen program of study at Harvard. What experiences led you to your research ambitions? Concisely state your past work in your intended field of study and in related fields. Briefly indicate your career objectives. Your statement should not exceed 1,000 words.

Here are some elements one might find in a good Statement of Purpose:

• a narrative explaining how one became interested in the subject in the first place (e.g. an applicant to a mechanical engineering program could explain how he became interested in mechanics as a child by watching his father do maintenence on his car)
• an account of one's education to date, which could include i) an account of how the decision to major in a particular subject was arrived at ii) descriptions of professors or other mentors who were particulalry influential iii) a description of the curriculum followed (particularly for international candidates)
• the envisioned area of study or research project to be carried out in the graduate program (note: it is not uncommon for people to change their minds about details of this statement or to change their mind about it altogether after admission - the important thing at this point is to show one's ability to formulate a reasonable plan)
• how gaining the graduate degree fits into one's overall career plans (e.g. a candidate for a Ph.D. in Economics could say that they hope ultimately to work in an international institution such as the International Monetary Fund or World Bank)