There is no short answer to your question.
The teacher who is responsible for a class might be called a “homeroom teacher” or a “class teacher” but different words are used around the world, because different countries have different systems of education.
In my high school in Western Australia, they were called “tutorial teachers”.
For the student, I would call him or her the “class leader” but I would not say he or she is “in charge” of the other students - most class leaders do not have that much power.
Student or Learner