Part II: Grades, Classes, and Institution of Origin
Part III: Letters of Recommendation
Part IV: Writing Sample
Part V: Statement of Purpose
Applying to PhD Programs in Philosophy Part VI: GRE Scores and Other Things
GRE scores tend to be required for U.S. programs, but they are less important than grades, letters, writing sample, and statement of purpose. Some schools don't even require them. According to this site, that currently includes Cornell, Emory, Illinois-Chicago, Michigan, Penn, and Wisconsin-Madison (among US PhD programs). [ETA: According to the comments below, Boston U. and Northwestern now also don't require the GRE.]
In my experience, some members of admissions committees take GRE scores seriously, others ignore them entirely, and still others employ minimum scores as a preliminary screen and otherwise disregard them. This will almost certainly vary by committee member, institution, year, and applicant details. (Foreign applicants, for example, might not be expected to have taken or done well on the GRE.)
Higher-level administrators play a role here: They oversee the admissions process, and in many schools they make the final decisions about fellowship funding. Since these administrators can really only evaluate your GPA, institution of origin, and GRE scores, students who do well on the GRE are more likely to get better funding offers than students with lower scores on the GRE -- for example, more years of fellowship without teaching (being paid simply to be a student!). Also, it looks good for the department if the students they admit have better average grades and GRE scores than the students in psychology, economics, etc. Since philosophy students on average do amazingly well on the GRE, even philosophers who don't think of GRE as diagnostic can find themselves citing students' GRE scores to make the case for financial support and for the superiority of philosophy over all other disciplines!
Therefore, I recommend that you practice for the GRE and retake it if your performance is disappointing. However, I don't recommend intensive training for the GRE. Devote that time, instead, to revising your writing sample and doing as well as possible in your classes and/or independent work.
Although averages will vary by school, my sense is that among students admitted to UC Riverside (currently ranked #32 in the U.S.), a typical GRE score is 160-167 verbal (86th-98th percentile) and 153-165 quantitative (51st-89th percentile), with totals in the 320-330 range. (No one I know takes the Writing score seriously, but 5 is a typical score.) Much lower would potentially be a disadvantage, whereas a nearly perfect score would be an advantage. Let me emphasize, however, that at UCR, and I believe most other places, a low score is not a defeater: Students with weak or (e.g., if foreign) no GREs are regularly admitted if their application is otherwise strong. Conversely, great GREs are at best a small favorable factor, more likely to help with fellowship opportunities than with admission itself.
There is no GRE subject test in philosophy.
[One philosopher taking a test]
Race, Gender, and Disability
Applications will often have optional tick-boxes in which you can indicate race/ethnicity, gender, veteran status, disability status, or membership in other social categories. Letter writers must also choose pronouns, and they might choose to mention disability or race if they think it's relevant. (Some would never mention such things. Others think they help the applicant by doing so, if the applicant is a member of a historically underrepresented group. If you prefer to keep the information confidential, tell your letter writers in advance.) Committees will often guess gender and ethnicity based on names.
Women and people of color are notoriously underrepresented in U.S. academic philosophy, compared to most other disciplines (data on other dimensions of diversity are harder to obtain). I believe there are persistent systemic biases. However, I also believe that most admissions committees would like to counter these biases and see a broader diversity in the field. Admissions committees may nonetheless show bias implicitly in how they read a file from "María Gonzalez" compared to a file from "Jake Miller", or in how they read a file from someone with a serious disability. For these reasons and others, it is perfectly reasonable not to want to disclose your race, gender, disability status, etc., to the extent these can be hidden. Don't let yourself be pressured into revealing something you're not comfortable revealing.
Schools that allow a "personal" statement in addition to a statement of purpose invite applicants to expand on obstacles they have overcome or other ways that they might contribute to the diversity of the graduate program. For discussion, see my advice on Statements of Purpose.
Presentations, Publications, Life Experience
If you have published a paper in an undergraduate journal or if you have presented at an undergraduate conference, or if you have other achievements of that sort, briefly mention it in your statement of purpose. However, they normally don't count for much.
If you have life experience relevant to your proposed area of study, also mention this in your statement of purpose -- but only do this if it is genuinely relevant, and err on the side of being brief and factual rather than overplaying it. For example, if you want to study philosophy of law and you have some work experience in law, mention it. If you want to study philosophy of race and you have worked with an organization focused on racial justice, briefly describe your experience and its relevance to your philosophical interests.
In disciplines other than philosophy, laboratory experience, work experience, and life experience are often an important part of the application. In philosophy, however, unless your situation is unusual, admissions decisions are almost always based on academic performance plus considerations of fit, balance, and diversity in the entering class, with other considerations having little weight.
Reapplying to Programs You Were Rejected from Last Year
Yes, this is fine! Likely, the admissions committee's composition will have partly changed, so you might get a fresh set of eyes. Also, hopefully, your application will be somewhat stronger.
... are sometimes accepted. This will vary by school.
Personal Contacts and Connections
Such things don't help much, I suspect, unless they bring substantive new information. If a professor at some point had a good, substantive, philosophical conversation with an applicant and mentions that to the committee, that might help a bit. But seeking out professors for such purposes could backfire if it seems like brown-nosing, or if the applicant seems immature, arrogant, or not particularly philosophically astute. Some professors may be very much swayed by personal connections, I suppose. I myself, however, often have a slightly negative feeling that I'm being "played" if someone who is applying to our PhD program contacts me during application season.
If you seek to build a personal connection with a professor, it's best to do so after application season is over or long before you have begun applications. The best way to build a connection is this: Carefully read something recently written by the professor (within the past four years maybe), then ask an interesting and well-informed question about it. You can send them the question by email or possibly ask them face to face at a conference or a local event. The odds of an email reply are probably below 50% and tend to be lower for the best-known faculty, who are inundated with emails from strangers. The chance of a sustained correspondence is even lower, but it's not unheard of.
Unless you are genuinely brimming with inspiration and enthusiasm, you probably won't want to attempt to build these kinds of connections as an undergraduate. However, I recommend remembering this advice for later. If and when you are an advanced graduate student, building connections in this way, outside of your home department, can be both intellectually rewarding and good for your career.
Applying to PhD Programs in Philosophy, Part VII: After You Hear Back