(http://www.career.cornell.edu/students/grad_prof.html)
Going to graduate school in an academic discipline usually implies entering a Ph.D. program and preparing for an academic career in that field. It requires background, typically a major, in the subject.
A master’s degree in an academic discipline (history or physics, for example) can be valuable for some careers -- teaching in secondary school is one instance. But a master’s degree has little value in certifying one as a practitioner in the field (as an historian or a physicist). One exception to this generalization is a master’s degree in one of the “area studies,” for example, Asian or Latin American studies. Often people preparing for a career in international service or business will “take” a master’s degree in the language and culture of the relevant part of the world. There are a number of what are called terminal professional degrees below the level of a doctorate.
Although majors are designed to prepare students for graduate work in that subject (and in this sense are “pre-professional” as well as the capstone to intellectual training in the liberal arts and sciences), many students decide quite late that they want to go to graduate school and even later what they want to study there. Interesting and successful students can be accepted for graduate study with the understanding that they will compensate for deficiencies in background. Sometimes, however, students need to take certain courses after graduation from an undergraduate institution before they are good candidates for admission to a graduate program.
Most students in Ph.D. programs receive funding from their institutions; parents are not expected to pay. A handful of extraordinary students win national fellowships like National Science Fellowships. Staff in Cornell Career Services and one of the college’s advising deans help students find out about and apply for these. But most qualified students receive a financial package along with the offer of admission from institutions that accept them. That package usually offers some combination of grant and salary for teaching.