There are two types of graduate qualifications offered in the US: a Master's degree and a Doctoral degree:
Academic Master's degrees - a Master of Arts (MA) or Master of Science (MSc) is awarded usually after two years of a combination of coursework and research. The coursework required can be between 30 and 60 credits, as well as a thesis and/or oral exam. These can lead directly to doctoral-level studies.
Professional Master's degrees - these degrees usually lead to a particular profession rather than doctoral studies. These include MBA (Master of Business Administration), MEd (Master of Education), MSW (Master of Social Work), MFA (Master of Fine Arts), etc. They usually require 36 to 48 credit hours of study and do not require a thesis.
A PhD is the most common doctoral degree, which requires students to produce an original piece of significant research, write a dissertation and successfully defend their work before a panel. The complete cycle of doctoral work may take anywhere between five to eight years. Professional doctoral degrees, including an EdD (Doctor of Education) or DBA (Doctor of Business Administration) involve a combination of coursework and research.
Undergraduate degree - you will usually be required to have completed an undergraduate (bachelor's) degree of four years in length to enter a master's programme.
Admissions tests - at Master's and Doctoral level, some departments may require you to take either the GRE (Graduate Recruitment Test) or GMAT (Graduate Management Admission Test). Other tests such as the MAT (Miller Analogy Test) for study in education or psychology may be required. Other subject-specific exams may be required to study medicine, law and dentistry, for example. Check with your institution as to which admission exam they require.
You will also have to prove you have sufficient English language ability by providing your TOEFL score or other approved English language test.
Read our guide to which standardised or admissions test you should take.
The process is very similar to that of undergraduate applications, though you may have slightly later dates for postgraduate application deadlines - confirm this with those universities you're interested in near the end of (or at the very beginning of) your senior year of undergraduate study.