A motivation letter is one of the essentials of your application to a US university. It is a unique opportunity of demonstrating a broader image of your personality to the university, not just providing a few figures and certificates. A good motivation letter for a foreign university is anything but a free-form essay on what a nice, outgoing and funny person you are. This abstract should deal with the essence of your life, accounting for the events which determined your future occupation, turned your world upside down and contributed to your choice of an American higher education.
A US university application form often includes two items in the list of documents: a Personal Statement and a Statement of Purpose. The former is mostly required for BA courses. Evidently, at this stage of your life you are quite unlikely to come up with a rigid plan for the nearest five or ten years, but you already have some achievements, aspirations and results to share. And these are exactly what you should include in your personal statement.
As to the Statement of Purpose, it is widely used in the application process of MA and post-graduate courses at American universities. In this letter, you are expected to give a clear picture of what you want to become and where you plan to be five or ten years away from now. You also need to prove that pursuing a course at the chosen university is an absolute prerequisite for achieving your goals in life.
An application letter to a US university with a request for funding. Most US universities consider your application for scholarship or funding by default. And this is when your essay comes forward, as funding is only granted to the best of the best.