The tutor who will teach your chosen subject will be assigned in consultation with senior colleagues in the subject concerned. Each arrangement is thus made on an individual basis taking into account the subject, the tutor's particular expertise.
There is, however, a wide variety of different kinds of tutors in Oxford, all of whom may be referred to collectively as "dons". (Don is Latin for teacher) At the top come the professors, about 8% of all teachers, whose title denotes not just that they are university professors, but that they hold a chair in the subject. The primary responsibility of a professor is to research and to lecture; apart from the occasional graduate supervision, they do not tutor at all, so you will not often meet them directly in tutorials. Where appropriate, however, the Advisers do persuade chairs to teach Associate Members.
Tutorial fellows form the backbone of the teaching system in Oxford; Fellows often hold university lectureships. The equivalent of fully tenured American professors, they are assisted by college lecturers (a misnomer, since they do not lecture at all). Research fellows (whether senior or junior) are engaged primarily in their own research, although they may take on a very few pupils for particular special subjects.
You may expect to be taught by tutors from any, or indeed all of these categories during your time at Oxford, since you will be taught by the same specialists, in the same way, and to the same standards as matriculated degree students. You may, of course, request particular tutors and every care will be taken to make appropriate arrangements where possible. However, no guarantee can be given that any particular tutor will be available to teach in any given term.
American students should also be aware that the PhD (or DPhil) is not regarded as important at Oxford. Many of the senior faculty (Fellows and Professors) never bothered with a PhD. They earned a First and wrote a book or books (the equivalent of a PhD) and were offered a Fellowship in international competition. Some of the most distinguished scholars have only earned a BA degree (the MA is honorary). Remember, it is a different system.
Oxford has always been a very international university...it was founded in large part by scholars from all over Europe (at that time every educated European spoke Latin). Today about 17% of the students are from overseas and about the same ratio applies to the faculty so you may have a tutorial from an Indian, an Australian, a German...or an American (about 5% of the faculty, including masters of three colleges, are American.)
The Academic Advisers in the colleges that we have been working with are all well aware that a OSAP student will only be in Oxford for a relatively short time. They will make a special effort, therefore, to see to it that these students get the most out of their time in Oxford.
Students accepted should know that they will be taught by the best teachers available in their subjects regardless of which college they themselves are affiliated. In the academic sense, therefore, it makes little difference with which college you yourself may be associated.
Changing tutors once arrangements have been made is only permitted in unusual circumstances, after consultation with your Academic Adviser. You should be aware that matriculated degree students are only rarely allowed to change tutors in the middle of the term. Given the close, one-to-one nature of the tutorial, it is inevitable that occasional personal problems arise. If you ever do have a problem of any kind with a tutorial you should talk to your Academic Adviser right away.
Your college may be thought of as your "home base"; you will probably find yourself doing work with dons from several other colleges.