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At MIT there are two kinds of advisors, academic advisors and thesis advisors.
Academic advisors are simple so we'll dispose of them first. Every graduate student is assigned a faculty member as academic advisor, generally in his or her area, though it depends on current advisor loads. The function of the academic advisor is to represent the department to you: to tell you what the official requirements are, to get on your case if you are late satisfying them, and to OK your class schedule. If all goes well, you only have to see your academic advisor in that capacity twice a year on registration day. On the other hand, if you are having difficulties, your academic advisor may be able to act as advocate for you, either in representing you to the department or in providing pointers to sources of assistance.
The thesis advisor is the person who supervises your research. Your choice of thesis advisor is the most important decision you'll make as a graduate student, more important than that of thesis topic area. To a significant extent, AI is learned by apprenticeship. There is a lot of informal knowledge both of technical aspects of the field and of the research process that is not published anywhere.
Many AI faculty members are quite eccentric people. The grad students likewise. The advisor-advisee relationship is necessarily personal, and your personality quirks and your advisor's must fit well enough that you can get work done together.
Different advisors have very different styles. Here are some parameters to consider.
How much direction do you want? Some advisors will hand you a well-defined thesis-sized problem, explain an approach, and tell you to get to work on it. If you get stuck, they'll tell you how to proceed. Other advisors are hands-off; they may give you no help in choosing a topic at all, but can be extremely useful to bounce ideas off of once you find one. You need to think about whether you work better independently or with structure.
How much contact do you want? Some advisors will meet with you weekly for a report on your progress. They may suggest papers to read and give you exercises and practice projects to work. Others you may not talk to more than twice a term.
How much pressure do you want? Some advisors will exert more than others.
How much emotional support do you want? Some can give more than others.
How seriously do you want to take your advisor? Most advisors will suggest thesis topics fairly regularly. Some can be depended on to produce suggestions that, if carried out diligently, will almost certainly produce an acceptable, if perhaps not very exciting thesis. Others throw out dozens of off-the-wall ideas, most of which will go nowhere, but one in ten of which, if pursued with vision, can result in ground-breaking work. If you choose such an advisor, you have to act as the filter.
What kind of research group does the advisor provide? Some professors create an environment in which all their students work together a lot, even if they are not all working on the same project. Many professors get together with their all their students for weekly or biweekly meetings. Will that be useful to you? Are the advisor's students people you get along with? Some students find that they construct important working relationships with students from other research groups instead.
Do you want to be working on a part of a larger project? Some professors divide up a big system to be built into pieces and assign pieces to individual students. That gives you a group of people that you can talk to about the problem as a whole.
Do you want cosupervision? Some thesis projects integrate several areas of AI, and you may want to form strong working relationships with two or more professors. Officially, you'll have just one thesis supervisor, but that doesn't have to reflect reality.
Is the advisor willing to supervise a thesis on a topic outside his main area of research? Whether or not you can work with him or her may be more important to both of you than what you are working on. Robotics faculty at MIT have supervised theses on qualitative physics and cognitive modeling; faculty in reasoning have supervised vision theses. But some faculty members are only willing to supervise theses on their own area of interest. This is often true of junior faculty members who are trying to build tenure cases; your work counts toward that.
Will the advisor fight the system for you? Some advisors can keep the department and other hostile entities off your back. The system works against certain sorts of students (notably women and eccentrics), so this can be very important.
Is the advisor willing and able to promote your work at conferences and the like? This is part of his or her job, and can make a big difference for your career.
The range of these parameters varies from school to school. MIT in general gives its students a lot more freedom than most schools can afford to.
Finding a thesis advisor is one of the most important priorities of your first year as a graduate student. You should have one by the end of the first year, or early in the second year at the latest. Here are some heuristics on how to proceed:
Read the Lab's research summary. It gives a page or so description of what each of the faculty and many of the graduate students are up to.
Read recent papers of any faculty member whose work seems at all interesting.
Talk to as many faculty members as you can during your first semester. Try to get a feel for what they are like, what they are interested in, and what their research and supervision styles are like.
Talk to grad students of prospective advisors and ask what working for him or her is like. Make sure you talk to more than one student who works with a particular advisor as each advisor has a large spectrum of working styles and levels of success in interaction with his or her students. You could be misled either way by a single data point. Talk to his or her first year advisees and his seventh year advisees too.
Most or all faculty member's research group meetings are open to new grad students, and they are a very good way of getting an idea of what working with them is like.
AI is unusual as a discipline in that much of the useful work is done by graduate students, not people with doctorates, who are often too busy being managers. This has a couple of consequences. One is that the fame of a faculty member, and consequently his tenure case, depends to a significant extent on the success of his students. This means that professors are highly motivated to get good students to work for them, and to provide useful direction and support to them. Another consequence is that, since to a large degree students' thesis directions are shaped by their advisors, the direction and growth of the field as a whole depends a great deal on what advisors graduate students pick.
After you've picked and advisor and decided what you want from him or her, make sure he or she knows. You advisor may hear ``I'd like to work with you'' as ``Please give me a narrowly specified project to do,'' or ``I've got stuff I'd like to do and I want you to sign it when I'm done,'' or something else. Don't let bad communication get you into a position of wasting a year either spinning your wheels when you wanted close direction or laboring under a topic that isn't the thing you had your heart set on.
Don't be fully dependent on your advisor for advice, wisdom, comments, and connections. Build your own network. You can probably find several people with different things to offer you, whether they're your official advisor or not. It's important to get a variety of people who will regularly review your work, because it's very easy to mislead yourself (and often your advisor as well) into thinking you are making progress when you are not, and so zoom off into outer space. The network can include graduate students and faculty at your own lab at others.
It is possible that you will encounter racist, sexist, heterosexist, or other harrassment in your relationships with other students, faculty members, or, most problematically, your advisor. If you do, get help. MIT's ODSA publishes a brochure called ``STOP Harrassment'' with advice and resources. The Computer Science Women's Report, available from the LCS document room, is also relevant.
Some students in the lab are only nominally supervised by a thesis advisor. This can work out well for people who are independent self-starters. It has the advantage that you have only your own neuroses to deal with, not your advisor's as well. But it's probably not a good idea to go this route until you've completed at least one supervised piece of work, and unless you are sure you can do without an advisor and have a solid support network.