<p>Basically it works like this:</p>
<p>1) The university, through its faculties and departments, organizes (Brit Eng "organises") lectures, seminar and practicals (Am Eng labs), administers exams, and grants degrees. The university, through its various faculty boards, also sets out the curriculum and the graduation requirements for each "course of study" (British equivalent of a major in the US). </p>
<p>2) The colleges, on the other hand, serve as student residences and also supplement university lectures with individual tutorials/supervisions given normally by teaching fellows who, in most cases, also have faculty appointments in the university. Furthermore, quite differently from what happens in the US, the colleges also handle undergraduate admissions. When you apply to Oxbridge, you have to choose both an intended course of study offered by the university ** and ** a particular college you wish to enter. Once you are accepted into a college, you are allowed to enroll (Brit Eng "enrol") as a university student. Most colleges accept students wishing to pursue any course of study (natural sciences, humanities, social sciences, law, medicine, arts, or engineering ), although a few colleges accept graduate (Brit Eng "postgraduate") students only. </p>
<p>3) Graduate students must also be members of a college, but unlike undergrads, they apply for admission through a university department/faculty and are assigned to a college "a posteriori". Sometimes, college-affiliated PhD students may also serve as tutors in college supervisions.</p>
<p>4) As I said before, the classes and exams one has to take to complete a certain course of study (Am Eng major) are determined by the university, not by the colleges. Therefore, the classes and exams you will have to take will not depend on which college you are in. All students in a given course typically take the same classes and the same final written exams (Brit Eng "papers") in their first year of studies. In the second year, the majority of classes and exams are still compulsory, while, in the third and (in the case of 4-year courses) fourth years, most classes are elective and students have the opportunity to specialize (Brit Eng also "specialise") in a sub-field within their course. It is also customary for third or fourth year students to complete a research project and/or write a graduation thesis.</p>
<p>5) As you've correctly said, colleges are independent bodies that normally own the buildings they occupy and have an endowment of their own. Some colleges therefore are richer than others, which may make a difference in terms of facilities and availability of student bursaries/scholarships (although, in practice, financial aid also tends to come directly from the university, not only from the colleges). Colleges also vary in size and, sometimes, even in gender balance (a few colleges still admit women only). Smaller colleges may have to hire outside tutors or have their student members go to other colleges for their supervisions/tutorials, but, one way or the other, all students are guaranteed the right to full access to the tutorial system (which is a major part of an Oxbridge undergraduate education).</p>