Personal statements, also called statements of purpose, are among the more important things you will write, as far as your career success is concerned. They are written as part of applications in competitive admissions for academic study programs. This includes admission to: college/university (note: In America, a "college" is a type of university; on what type of post-secondary institution to apply to, see our essay "What kind of college or university should I go to?" on this site), art school, graduate schools (for MA and PhD programs), professional schools (law, medicine, business, divinity, certain arts), medical residencies, and other study or training programs that have competitive admissions.
These applications always have several parts. For instance, in graduate school applications, the elements are undergraduate grades and grade point averages both overall and in the major or relevant field of study; GRE scores; letters from faculty or other persons who know you well from having worked with you in some way; and your personal statement. Programs vary in how much weight they put on these four elements. In some cases, the personal statement is the most important thing. Applications to undergraduate study typically contain several parts that require written answers from you. But the statement is always the central document that says in your voice who you are, what you want to do, and what you have to offer.
Personal statements for undergraduate admissions and those written for graduate and professional programs are very different. The undergraduate application essay, which this is also called, allows for great creativity; they are not so interested in what you propose to study and how, because it is expected that undergraduates at American colleges and universities discover this while there, usually in the first two years.
Note that at the more prestigious American colleges and universities, being an undergraduate student is an experience of personal development in a kind of extended post-adolescence whose completion is marked by calling graduation "commencement": it inaugurates the young person's life as an adult entering some profession, no longer primarily concerned with learning; this experience, known as the "liberal arts education," with deferment of choice of major course of study until the beginning of the third year, is also an experience of socialization into the elite professional class, which may be a reason why universities have the become focus of anxieties centered around social groups historically excluded from the elite in a time of economic crisis. This privileged socialization has real personal as well as professional advantages, whether or not it is worth the price. To compete successfully in this environment, you need to give it your all, and, whatever your background, the personal statement is a key element of getting in and making it.
In a college admissions essay, what they want is for you to tell them who you are, and how you think. Showing maturity has value. You may wish to take as a model the American literary essay. Typically, it describes a personal experience that one learned from. Writing well is also of great value here. Some people have famously gotten admitted to very prestigious universities while blatantly flaunting in their writing the usual norms, just be submitting an essay that seems brilliant.
Graduate and professional school personal statements are very different. These are concise (1-2 pages) statements of what you want to do in graduate school and why and how.
Occasionally, a writer is able to make strategic use of a relevant personal anecdote. One of the most successful PhD program statements of purpose I have seen started with an account of a barbaric crime that happened in the applicant's home country to someone she was close to. This crime was of a type that is all too common there. She mentioned it, starting the essay with it, because she was in an area studies field and hoping to work on this problem or ones related to it. That first paragraph was written with such compelling, passionate, tangible prose that it, and the use to which she put it, made that statement so compelling that a reader might well have wondered not, shall we admit her, but how can we not admit her? She got her first choice program, also the best in her field. Most people cannot do this, quite, but this example points to something fundamental that all applicants share: Though in graduate and professional schools, as in the professions they prepare people for, it tends to be taken for granted that the writer cares, but any opportunity to indicate a deep and passionate commitment that will seem genuine and not contrived should be seized upon and expressed.
What is needed in all cases is to say something about why one has chosen this field, what you would particularly like to do, and why the program is a good match for you. In personal statements for PhD programs, something else is needed, and it is key. Give some indication of the area of a possible dissertation project if you can, and mention some (more than one; 3 may be an ideal number) faculty members whom you would like to be able to work with. That will mean, ultimately, who could be on your dissertation committee. Departments look to match applicants to faculty, this plays a role in admission decisions, and they can try to figure this out or you can help them. What you say about the faculty members should be informed. Of course, this part of the essay and any that is specific to the school in question will have to be redone for each school to which you apply. (The rest of the statement can be the same). The department will list its faculty on its webpage with keywords or a paragraph or more on their interests. Naming their interests, though, is not enough, except in weeding out departments to apply or not apply to. Use Google, and consider using a scholarly database (such as JSTOR in the humanities, which makes articles available for reading to anyone for $20/month) to find out more about them. Reviews of their books by other scholars may help, or you may want to read some articles, or even visit the library and skim through and read a bit of one or more of their books. You can use this research to write one or two sentences on each professor you choose to mention. E.g.: You are interested in Smith's work on X because of your interest in Y. The art here will be in writing your 1-2 sentence interpretation of their work or some part of it. This also shows some skill at a kind of thinking you will be doing much of, especially if you are in a humanities field. This is not needed for master's or professional programs that do not involve a doctoral dissertation; in that case, it is enough to reference features of the department you like, and maybe certain faculty more broadly, as they will not be holding your hand so tightly as they do in PhD programs, where you really are their apprentice. For the same reason, you should give a good indication of what you want to do. Optionally, you can present a possible dissertation topic; in that case, what's most important is both that it's the type of topic they might like, and that you show you can think out clearly a plausible research topic; they know you can change it later.