The BMS Phase I is a basic graduate course phase while Phase II is the PhD research phase.

Phase I

The typical entrance is with a bachelor's degree. Each student is registered as regular student at one of the three universities and has a mentor at the same university.
Phase I takes usually 3-4 semesters to complete. It requires the students to successfully complete five Basic Courses and at least two Advanced Courses (s. below), including a seminar. The BMS Basic Course program is held in English and coordinated between the three universities. Students are expected to attend the BMS Friday Colloquia regularly.
At the end of Phase I students have to pass the oral Qualifying exam in order to continue into Phase II. Phase I students must also use the Phase I to find an advisor for their dissertation research in Phase II. Attending more than the one mandatory seminar is a good way to get to know professors and their research, to find out what the open questions in the field are and if the professor is willing to take on more Ph.D. students.
The BMS study program for Phase I includes two one-semester Basic Courses in each of the seven Areas.
You can find descriptions of the Basic Courses here.

Advanced Courses: Besides the Basic Courses, the BMS offers Advanced Courses in each of the Areas. Their aim is to bring students in a specific area to a level where they can engage in independent supervised research. Phase I students are expected to take at least two Advanced Courses, of which one must be a seminar course. The frequency with which Advanced Courses are taught and their specific contents vary. In most cases, they are self-contained one-semester courses (4 hours/week). At least one Advanced Course in each area is offered every semester, and each year at least one of them will be a seminar course. Advanced Courses are usually organized in close collaboration with one of the research training units associated with the BMS.

Phase II (Ph.D. research phase)

The typical entrance is with the Qualifying Exam, or a master's/"Diplom" degree. Each student is registered as a PhD student in mathematics at one of the three universities. He/she has an advisor at the same university, and an additional mentor.
Students are expected to finish their dissertation, i.e. Phase II, in 4-6 semesters. They should conduct specialized research on their thesis in one the Berlin mathematics research groups, attend at least one advanced course per semester (for the first two years), as well as the BMS Friday Colloquia.
Completion requirements are set according to the PhD regulations at the respective university. The PhD regulation of the three universities can be found here.
In addition, the BMS awards the "Certificate of Excellence" for the completion of BMS program.

BMS study regulations NEW
Old BMS study regulations (optional for students who entered the BMS before April 2015)

The PhD requirements of the three universities are available for download here:
Guidelines and Regulations