Whatever they are called, they are essentially essays which are written in response to a question or questions on a graduate or professional school application form which asks for some sort of sustained response.
Some applications ask more specific questions than others.
There is no set formula to follow in shaping your response, only choices for you to make, such as whether you should write an essay that is more autobiographically focused or one that is more professionally focused. From application to application, requested personal statements also vary widely in length, ranging from a couple of paragraphs to a series of essays of a page or so each. It's most likely that your personal statement will be read by professors who serve on an admissions committee in the department to which you are applying. It is important in developing your personal statement to carefully consider this audience. Obviously, it is important that personal statements are not simply prose formulations of material contained elsewhere in the application.