In China, the teacher in charge of a class is called "Bang zhu ren", who not just teach, but also take care of students' life beyond study. So in English, what's the expression?
Mentors and faculty advisors are the terms used. Both of them can be in charge of a class or a group of students.
Faculty advisors are teachers whom you can approach in case of any difficulty like academic registration, help with choosing subjects and for obtaining permissions. You can find more information about their duties here.
A mentor maintains a more personal relationship. S/he is supposed to bond closely by going out on short trip and host dinners. Mentors act as a guide to choose your area of interest in your specialization, counsel you when needed and literally be your "on-campus parent".
class teacher (Macmillan)
a teacher who teaches or is responsible for a particular group of students in a school