As I've already saw, many people here are related to admission process. And I want to ask their opinion on what is priority of all stuff from application: what is the most important (letters of recommendation or personal statement, GPA or maybe test scores), what is less important for admission and what is the least important? Would admission committee prefer applicant with good scores, high GPA, but with not outstanding personal statement and without letters from famous or well-known professors (for example, my scientific advisor in undergraduate school get his PhD only a couple years ago), or student with less GPA, test scores, but with letter from university's prof?
Some graduate schools require General GRE (Graduate Record Examinations). Would it better to send GRE scores (general or subject or both) even if it's not required (in case of relatively good result), or them wouldn't be considered at all?
If we're talking about a PhD candidate, the most relevant factors, for me, are those that demonstrate the applicant's capability to be a productive researcher. That means a compelling statement of purpose (why the applicant wants to be a PhD student, with ideas for potential areas of exploration, if not necessarily an entirely thought-out project) counts for quite a bit.
Equally valuable are letters of recommendation that actually talk about the candidate. I don't care for a letter that regurgitates the student's academic performance, or that reads like a template in which the name of the applicant was swapped in for somebody else's. That does me no good in evaluating a candidate, and to some extent can work as a mark against a candidate, since it indicates a lapse in decision-making (why get a letter from someone who doesn't know you well?). A really good letter of recommendation, however, can sometimes make the difference between somebody who's on the bubble and somebody who gets an offer of admission.
Transcripts and test scores do matter, but for me are less valuable, because they don't give me a lot to go on—it's not easy to tell what's a "good" performance and what isn't, particularly when schools use "nonstandard" scales (such as reporting scores out of 100 without telling me what the average score is!).
The other factor that we do here that not all programs do is an interview (in person or via Skype). This is perhaps the most important part of the process for us, because we can see if the paper record matches the "actual" applicant. Sometimes candidates look good in writing, but can't really talk knowledgeably about what they have studied. That's usually a clear sign that someone really won't be a good fit as a graduate student.