We used Common App for St Andrews, but their supplemental question is basically supposed to be the same as a UCAS personal statement, except you can further tailor it to St Andrews specifically.
In terms of gathering information, we:
(1) Talked to the St Andrews rep at a college fair;
(2) Looked at advice articles and successful examples online;
(3) Read the St Andrews guidance available online.
Really that last was the most informative, so here it is:
https://www.st-andrews.ac.uk/study/apply/ug/
Personal statement
Your personal statement provides you with an invaluable opportunity to tell us your personal reasons for studying your chosen subject at the University of St Andrews. This is a very important part of your application. Most of the students who apply to the University will be well qualified, so decisions on who will receive offers will often be determined by the quality of the personal statement. You are encouraged to spend time drafting and re-writing your statement so that it is organised and well-written.
It is a good idea to give some relevant background information about yourself, your interests, and especially your reasons for your choice of course. You might want to highlight any relevant work experience or voluntary work you have done. If you have an idea of what your career choice might be, and how your studies at St Andrews would support that, you may wish to include this as well.
In summary, your personal statement should give the admissions officers a picture of you: someone who is interested in the subject area for which you have applied and who has the motivation and potential to do well in a university environment.
In terms of the second thing, we found the library of personal statements at The Student Room most useful to narrow in on personal statements used for our course (Biology, in our case):
https://www.■■■■■■■■■■■■■■.co.uk/university/personal-statements/
There are also a lot of advice articles linked there. This one, for example, covers a lot of issues:
But in a lot of ways, they are just echoing that St Andrews advice. Indeed, here is another article from admissions insiders, but it largely is just duplicative:
My take away was basically you just have to imagine the admissions tutors for that course asking themselves, “Do we want this student in this course at our university?” It is not really that complicated of a task from that perspective. And they are primarily looking for a meaningful explanation of your interest in, and aptitude for, that course. If you have directly relevant activities, work experience, and career goals (that last mostly just for “vocational” courses like medicine or law), you can include those too.
Do that in a well-organized, detailed fashion, and I gather that is going to be the core of an effective personal statement.