I'm not sure if this is the case for all postgraduate programs, but my research masters course had a 'course tutor' who was basically in charge of the course and the students on it. He was our first point of contact for anything, basically. He was the guy who interviewed us for the course (if we chose to visit the department in person), he sorted out our offices and computers, he introduced us to the course content and structure etc.
Once we were settled in, he was there to basically advise us, help us find projects and supervisors, help with any issues we had, academic or otherwise, etc. and he also taught one of our classes.
The thing was, he actually ended up being more than just an academic tutor to us. Being fairly young, only a few years older than most of us, we were on first-name terms with him. He invited us round to his house on occasion (we painted his 3-year-old son's bedroom for him
) and we got to know his wife and son. He was always hanging around in our office, joking around and basically doing nothing that resembled research, lol. He would come out drinking with us and attend birthday meals, and we're all friends with him on Facebook and MSN (he even came to my goodbye party before I left for the US in January)!
All around, a pretty cool guy - apart from the lacking organisational skills (the course involved a month-long field trip to Central America which we ended up organising and booking ourselves because we would never have made it there if we'd left it up to him
).
In contrast, for undergrad courses, the tutor-student relationship is more formal. You have weekly 'tutorials' with the professor in groups of about 4 students, and you discuss questions about your lectures or homework assignments. You can go to them for any other help too (personal/academic), but I don't think many students actually bother to (I had two tutors, both male, both Physicists - one was Bulgarian and the other was Russian - so they weren't that easy to approach about personal issues)!