As a graduate student, the whole process of student-faculty matching was such an enigma. I had visions of faculty in a dark room staring at student pictures, arguing over who got who, blatantly ignoring the student's preferences.
In reality, nothing could be further from that vision. And contrary to what I thought, the process is different every year. The only constant is that the process is a cluster-f***. Here is how it generally works:
1. Students meet with faculty.
2. Students indicate their top choices.
3. The graduate coordinator goes to the faculty member and says, "Hey, student X put you down as #1. Do you want them?"
4. The faculty member says yes or no.
5. Discarded students (and discarded faculty) then go through another round of matching. Believe it or not, there are some faculty (discarded faculty) that no students indicate a preference for. This can be because the faculty member has a bad reputation, has a boring project, or is super-new. Discarded students are those that didn't get their first choice and things just didn't work out. It often does not reflect badly on the student - it is just a matter of bad luck. Sometimes it isn't, where the student clearly isn't going to pass quals, so the faculty don't want them. But students, I assure you that most of the time it is just bad bad luck.
6. And then the politics. And that's where the sh**-storm commences. Super-senior-famous faculty might try to pluck a student from a previously happy faculty student-match. (This is happening to me now). Or your department head might demand that you take student X because no one else wanted them. Or we have students that demand to pursue topic X, even though no one doing topic X has funding that year. Else, we have faculty that decide to take students from other depts, which disrupts the balance of openings to students internally. When people start meddling - and they do every year - it gets crazy.
So that's why it takes so long for the faculty-student matching process. And that is why it is different and bizarre every year. Most of the time, I am the #1 choice for the students I get. Other times, I am the #2 choice. I just want the best people and I usually get them. Most of the time it all works out. But damn it's a mess.