Every major American university has a political science graduate program awarding a Ph.D. degree, and international relations is a field within almost all these departments. The departments also award a Masters of Arts degree, but this is not particularly useful. The Ph.D. requires two to three years of coursework, followed by comprehensive examinations and a thesis, which usually takes another year or two full-time; obviously the time will be longer if students have to go part-time because of limited resources.
The Ph.D. degree is basically a research degree. It is essential for anyone who wants to teach at a college or university, and it is often found among researchers and analysts working for the government as well. On the other hand, the degree is given in political science; students can concentrate in international relations, but are required to take courses and examinations in other fields such as American government and political theory as well. Moreover, it takes longer to get than any other option discussed here, and it is not clear that it is worth the extra effort and money unless you are going into college teaching. One group of Foreign Service examiners, when asked how useful graduate school would be, said that the two to three years of coursework would be useful, but that the candidate would do better spending a couple of years in the Foreign Service than working on a doctoral dissertation. Several programs retraining people with Ph.D. degrees to go into business have been fairly successful, but this is a pretty roundabout way to get into business; if that is the desire, try for graduate business school.
A major in political science is not required for admission to graduate programs in the discipline, but students should take at least enough courses to decide if they want to do this full-time for a long time; the biggest difference between graduate and undergraduate work is that they must live one subject twenty-four hours a day. Admission is usually based on grade point average, Graduate Record Examination (yet another standardized test) scores, and faculty recommendations. In general the best departments are found in the leading universities. If students want more specific guidance, talk to international relations faculty; this is one subject they know something about, and they will be current on the varying reputations of political science programs around the country.