Graduate (i.e. Master’s) programs are the second stage of higher education. These advanced degrees are typically more focused than undergraduate ones. While undergraduate students could choose between wide array of subject and classes, the choice for graduate students is more limited: the number of options may be significant,but all classes are closely related to the main focus of your program. Graduate program is usually a way to deepen your knowledge in a narrower field or to gain some practical skills for your future career.
In the United States there are two main types of graduate programs: Master’s programs are usually professional degree which prepare students to the career in business or public sector. The most prominent examples are programs provided by law or business schools (but many other examples include public policy schools, schools of education, public health etc.). The second type of Master's programs are usually just a part of PhD: during the first two years students attend classes, get a Master's degree and then start to work on their own research projects. However, it's ofter impossible to apply to the Master's programs of this kind directly. That is why is’s quite difficult to find a Master’r program in Physics or Mathematics in the USA.
The situation in Europe is slightly different: Master’s programs exist for those who want to pursue a career in academia as well as for those who want to focus on career in business or public sector. In the first case Master’s program would be a first step on the way to PhD, but not an integral part of doctoral program. Students could always change the university after completing a Bachelor's or Master's degree in order to get more experience and enlarge their social network.
In order to be admitted to the graduate program of their choice, students usually have to provide undergraduate transcripts, letters of recommendations from their professors or supervisors, as well as the motivation letter (or statement of purpose) which demonstrate the reason why students choose this particular program.