Priority of application materials for admission decision 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?
As aeismail says, what we are looking for is concrete evidence of research potential.  So, in decreasing order of importance:


Decent grades and (if required) test scores; otherwise, no one will read your application at all.  Here, "decent grades" means three-point-something from a good undergraduate program.
Research publications, if you have any.
Recommendation letters.  You must have at least one (and preferably three) strong recommendation letter from a faculty member who praises your research potential in specific and credible detail.  Letters that draw specific comparisons to other successful PhD students are best.  Letters from junior faculty are perfectly fine; they can draw comparisons to their recent graduate school peers.  Letters that say only "He got an A+ in my class" are useless; we can read your transcript.
Research statement.  Your statement must discuss your research experience and interests in specific and credible detail.  A statement that only describes your sources of inspiration ("Ever since man walked on the moon...") and/or brags about coursework is useless.
Other concrete evidence of independent research/scholarship/creativity.
No red flags.  Potential red flags include low grades in classes central to your proposed research area, missing key classes entirely, abysmal test scores, negative (or overly delicate) recommendation letters, recommendation letters obviously written by the applicant, spelling and grammar mistakes in the research statement, any evidence of immaturity or personality issues, and any evidence that the applicant is not prepared/informed/serious about research.


Notice what's not listed.