You didn't say what kind of graduate program you are talking about. The admissions criteria for a Master's program are very different than for a PhD.
Here's what the top-three admissions criteria look like for admission to a research-oriented PhD program:
- Evidence of research ability.
- Evidence of research ability.
- Evidence of research ability.
So, if evidence of research ability is so important, how is it judged? Well, there are several ways that applicants can demonstrate research ability:
Demonstrate prior success at research. For example, participated in one or more prior research projects that led to a publication at a peer-reviewed place. This is usually the strongest evidence.
Show prior experience with research, with evidence that it went well or that future research will likely be a success. For example, participated in one or more prior research projects, which did not lead to a publication, but the letters of reference state have positive things to say about the applicant's research ability, and the letter-writers are credible on this. This is next-best.
Show great intelligence and technical ability, as well as passion/motivation. Here we are talking about indirect measures of research ability. One of the strongest ways is to excel in technical classes. Admissions committees will also look at the motivation/drive (what does the applicant want to study? why? is the applicant driven to do research? why?), at written and other communication skills, and other factors.
Of the materials in the application packet, I could prioritize which are most important:
Publications. If you have publications, include them. Admissions committees will often read the publications, look to see where they are published, etc.
Letters of reference. Strong letters of recommendation can be very valuable. They need to come from credible people who are well-calibrated about what it takes to be successful in a Ph.D. program, and they should be as strong as possible about the applicant's research potential and other abilities.
Classes. Great grades in relevant courses is helpful. The courses also need to provide adequate preparation for the Ph.D. program.
Essays. The personal essays should be thoughtful, well-written, demonstrate the applicant's interest in research and goals for Ph.D. study. Admissions committees will read to see whether the essays seem well-informed about the field the applicant wants to join. They'll also try to figure out the applicant's most likely interests, to see if they are a good match for the faculty in the department who are looking to advise new Ph.D. students.
Other materials. The rest of your application packages (e.g., GRE scores) are of lesser importance. They're more likely to get you rejected, or raise a red flag that causes the committee to look more closely at the rest of your application packet, than they are get you accepted. It is semi-important to demonstrate communication skills; if you cannot communicate clearly in the language of instruction at the university, then you may not be able to serve as a teaching assistant, which means the school may not have a way to fund you, which is very bad. Also, advisors are more likely to want to work with someone who has good written English than someone who will need to learn how to write clearly.