Let me take a crack at the question, since I am currently on the graduate committee at UGA. [The University of Georgia is about the 50th best department in the country, so just a little shy of being a research 1 university. We are strongest in algebra, number theory, and algebraic geometry and are able to attract some excellent students in these areas.]
The two most important things for us are:
1) Very good to excellent grades, in courses which go beyond the minimum necessary for a math major and include, if possible, at least one graduate course. We are looking for all grades B or higher and at least as many A's as B's (which implies a GPA of at least 3.5). One or two poor grades will not concern us too much if they are in lower level courses, are followed by several years of better grades, or some explanation is given in the personal statement and/or the letters of recommmendation. A successful applicant will probably have taken real analysis, abstract algebra and topology. We certainly do take into consideration the student's school: e.g. a student from a liberal arts college may not have any graduate courses available.
2) High GRE scores. We like to see at least 700 on the GRE quantitative. It is not as criticial, but I would like to (and often do not!) see at least 600 on the GRE verbal; both scores are kept in mind by the university when it makes decisions about who will get prestigious fellowships. As for the GRE math subject exam: I am sorry to say that as of this year we do not require it. After looking at other universities of equal and greater status, we have decided to start requiring this exam next year (i.e., for students who are applying to start in Fall 2011), although we recognize that this may shrink our applicant pool. A score in the top 50% on the math subject exam looks good to us.
Next come the recommendation letters, which we use to gauge the student's enthusiasm, ability and preparedness for graduate school as compared to other aspiring graduate students. It is much better for us if the letters come from someone that we have heard of, or whom we can verify is a successful research mathematician (I have looked some recommenders up on MathSciNet). Good things to see in such letters are comparisons to other students who have gone on to be successful at research 1 graduate programs.
REU experience looks good, especially if accompanied by a recommendation letter from the REU supervisor who can be specific about what is accomplished. Sometimes students do enclose papers or preprints that are the result of REU work. Again we like this in general (more if the paper looks interesting, less if it looks rather trivial) and may forward this along to other faculty members to see if they are especially interested in the student.
I would say that the personal statement is in fact not very important, except perhaps to address/explain weaknesses in other parts of the application. [I accept that it might be more useful at a different institution. I have also advised graduate students and postdocs applying for academic jobs that their cover letter is not very important, and I know that some people -- especially at liberal arts colleges -- have said exactly the opposite.] It is useful as a writing sample, and a lack of spelling, grammatical and punctutation errors is evidence that the student is serious about their application.
In fact it is probably more likely that you will lose points in your personal statement than gain them. When I was a college senior, we had a Q&A session about applying to grad school. The head of the computer science department (Lance Fortnow, I think) told us the following story: he once had an application from a candidate who had very strong grades, GRE scores and recommendation letters. But in his personal statement he was asked "Why do you want to go to graduate school in computer science?" The candidate's response "Because I am trying to avoid working very hard" was exactly the opposite of what the department head wanted to hear. The rest of the application was so strong that, albeit with some misgivings, this candidate was admitted. The result was disastrous: the candidate really didn't want to do any work so was (of course!) a most unsuccessful graduate student, eventually getting kicked out of the program. The CS department head concluded that after this experience, he would never admit a candidate who said something like that on their personal statement. (And neither would I.)
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