As you have asked about the Stanford University requirements to get admission in PhD in USA, following things are necessary

Application Requirements:
Application Form:

Complete and submit Stanford's graduate online application.
Application Fee:
Application fees should be submitted with the online application.

Statement of Purpose:
Your statement should be typed, single-spaced and should be between 1 to 2 pages. Describe concisely your reasons for applying to the proposed program, your preparation for this field of study and why program is a good for you, your future career plans, and other aspects of your background as well as interests which may aid the admissions committee in evaluating your aptitude and motivation for graduate study.
Resume:
Resumes are required of all candidates. Upload your resume in the online application.

3 Letters of Recommendation:
Candidates are required to get 3letters of recommendation. During the online application, you will be asked to identify your recommenders and their email addresses.
Recommendations should be written by people who have supervised you in an academic, employment, or community service setting. At least one should be from a university professor familiar with your academic work.

College and University Transcripts:
When submitting your online application, upload your transcripts as a PDF.
The instructions for sending transcripts on the online application and on the general Graduate Admissions Office website differ from this Stanford GSE requirement. Transcripts should be submitted for any institution at which you spent at least one-year doing academic study.
Concerning course work completed in a study abroad program:
If the course work and grades are revealed on the transcript of your home institution, you do not need to submit original transcripts from the study abroad institution.
GRE:
The GSE strongly encourages all candidates to take the GRE General Test before submitting the application and by October 1. Candidates are required to take the GRE General Test before the application deadline and within five years of the application submission date.

TOEFL
Stanford University requires the Test of English as a Foreign Language (TOEFL) from all candidates whose native language is not English. The Test of Written English (TWE) portion of the TOEFL is not required. Applicants who have completed a four-year bachelor's degree or a two-year master's program (or its equivalent) in the U.S. or at an institution where English is the main language of instruction are not required to take the TOEFL
The Graduate School of Education requires a least possible TOEFL score of 250 for the computer-based test, 600 for the paper-based test or 100 for the internet-based test in order to be measured for admission.