Like the SAT, the GRE general exam tests people in both verbal and quantitative reasoning. While your SAT scores were required to get into an undergraduate program, though, the GRE is generally required to get into a Ph.D. program. Not every program requires that you take this test, though most do. Some programs may require additional standardized testing, but in any case, you will likely have to take the GRE.
Some Ph.D. programs require prospective students to first earn a master's degree, which typically takes about two years. This is not, however, a universal requirement--some Ph.D. programs require only that you have earned an undergraduate degree. This is also true for MA/Ph.D. programs, which are joint programs in which students earn their master's degrees and Ph.D.s at the same time. Whatever degree is required of you, a good academic record is required for virtually any program.
Before admitting a student, a Ph.D. program must determine if she produces work of the right quality. For this reason, students must submit writing samples as a prerequisite for admission into Ph.D. programs. Thesis papers from master's degree programs are particularly useful samples because they demonstrate the ability to write at a graduate level while also showing the student's area of interest.
In many Ph.D. programs, students develop close working relationships with their professors. That in mind, professors need to be sure that potential students are effective learners and put forth the kind of effort that will be required of them. It is for this reason that most Ph.D. programs require letters of recommendation from previous teachers or other authorities, like internship supervisors, that explain why the students should be considered for admission.