A Breakdown of the Trifecta
When you attend lectures, you will go to your department and be in classes or labs with students who are members of different Colleges from all over the University. You will live and maybe eat at your College while you attend the University, generally speaking.
The Colleges
Colleges are privately endowed, independent bodies that provide accommodation, support, academic help (through supervisions - though most supervisions are done through the department for third years), and a community for students at Cambridge. Each of the 31 Colleges has its own grounds and facilities, including a dining hall, chapel, library, bar, computer lab(s), and JCR (Junior Combination Room, which is a lounge where undergrads can "combine" with each other and watch TV, read papers, sometimes play pool / snooker or foosball. JCR also stands for the undergraduate student government at each college). All students belong to one College, where they matriculate at the beginning of their first year. In addition to students, Colleges also have senior members, who also work in one of the University's departments. Some of these senior members are fellows. Each College has its own administration and each College charges its students fees, namely for housing, food, and general membership.
Tutors
One of the main roles of the College is to provide support, both welfare and academic. In all Colleges, each student has a Tutor (shared among many students) with whom they meet once a term to see how things are going, and to whom they can go if there are any problems. The tutor is something like an advisor (at MIT) who has nothing to do with academics, but rather with your general welfare. The Dean of Chapel is also available for consultations (religious or welfare). In any case, you have to ask for help or consultation if you need it; except for the meetings with your Tutor, people won't usually seek you out to see if you need help. If you have any general questions or requests regarding matters in your College (e.g. you want to find a guest room for visiting family or friends), your Tutor is a good place to start.
Senior Tutors
Above your Tutor is the Senior Tutor, who is responsible for the welfare of all students at the College and also runs the day-to-day operations of the College.
Master
The formal head of the College is the Master, though his/her role is much more broad and less concerned with the details of College operations. The Bursar is the head of the College's finances. It is to his/her office that you will go to pay your fees or to ask any questions you have about these College fees.
Porters
Another very important part of the College community and organization are the porters. It is hard to put all of their jobs into one description. Porters are responsible for College security, sorting mail or holding packages, holding keys to various College facilities, and basically helping you anyway they can. Most Porters are very friendly and are a great place to go for any question you may have; if they can't give you an answer, they'll point you to someone who can. They are can be found in the Porters' Lodge, which is usually located immediately adjacent to the College's main entrance.
The University
The University's main purpose of is to "examine and confer degrees" and it is responsible for overall administration. But it is hard to put a finger on exactly what the "University" is and what it does, because most of the students' dealings are with departments (faculties) and Colleges. Generally speaking, the University is the highest level of organization at Cambridge. If you play a sport at a very high level, you may wish to try out for the University team. Likewise, dedicated musicians, debaters, actors, etc. will find people from all Colleges in the high-caliber University societies. Technically, the Chancellor sits at the head of the University, though the role is really just ceremonial—it is held by Prince Phillip, the husband of Queen Elizabeth. The Vice-Chancellor, much like the President of MIT, runs of the affairs of the University.
The Departments
The departments (or faculties) are on a lower level of organization than the University. They provide homes for all the education and research that is conducted in the University. The departments report to the University, which is therefore centrally responsible for all of the education that occurs at Cambridge; the students pay the University tuition fees, not department fees. (Also, it is the University that administers exams at the end of each year.) Each department has its own building (or set of buildings), administration, and teaching and research staff. However, many lecturers of the departments (and therefore also of the University) also fill roles as senior members of the Colleges, thus further muddling the distinction. Professors in departments at Cambridge are comparable to full professors at MIT, only they are rarer at Cambridge. Lecturers and readers are the equivalent to assistant professors at MIT. Your supervisors, who conduct your supervisions (discussed below in the section 5.3), are professors, lecturers, post-docs, or graduate students who are members of your department
Director of Studies
Your main source for academic support is your Director of Studies (DoS), who is a member of your department (and usually a member of your College, unless your College doesn't have any qualified senior members in your department). His or her role, similar to that of an advisor at MIT, is to look after your academic welfare, advise you on subject choices (within your department), monitor your progress, appoint you supervisors (for your supervisions), and deal with any academic problems that may arise. If you chose to take classes outside of your major (e.g. HASS courses), you will need a DoS for that discipline. This DoS may be assigned to you or you may have to actively seek him/her out (because cross-disciplinary study is not normal at Cambridge). It is very important that you be proactive about meeting with your DoS(s) at the beginning of each of the three terms, especially if you need special arrangements (possibly for extra supervisions) at the beginning of the year.