What is described above is one manifestation of the GRE, the General GRE. The three sections, Verbal, Analytical, and Quantitative, are scored out of 800, and the score is determined by the computer using some complex algorithm based on the difficulty level of the questions, which themselves were determined using an algorithm based on how the test taker answered the first couple of questions. It all seems a bit convoluted to me.
In any case, the real purpose of these sections are as follows:
Analytical: Thinly veiled IQ test
Verbal: Thinly veiled test of socioeconomic class
Quantitative: Thinly veiled test of how many times you skipped class in high school.
The other manifestation is the so called Advanced GREs which are the subject tests. They are supposed to assess undergraduate performance and preparedness for graduate school in the field in question. There were formerly subject tests in most of the sciences, many of the social sciences, and most of the humanities, but over the past few years most, especially in the humanities, have been discontinued, presumably due to lack of interest on the part of both graduate admissions committees and applicants. Advanced GRE exams remain for most of the sciences, including Physics, Chemistry, Biology, Biochemistry, Computer Science, and Mathematics, as well as Economics and possibly a few other subjects, and in most of these subjects they are mandatory or at least highly suggested for graduate admission.
The Advanced GREs are notoriously hard, but I found, at least in dealing with the Physics one, that the key was achieving familiarity with the test and the types of questions on the test. Once this familiarity was cultivated, many questions could be answered by process of elimination of the answer choices and without actually solving them explicitly, which of course would be impossible anyway given the time constraints.