Is there a word in English for a student in a classroom, typically in elementary, middle, or high school, who would be in charge of keeping order in the absence of the teacher? This person is to tell the teacher, or someone higher, the names of his/her classmates who misbehaved and might put their names up on the board for the teacher to see.
I encountered the terms "class monitor" or just "monitor":
a student appointed to assist in the conduct of a class or school, as to help take attendance or keep order.
a person appointed to supervise students, applicants, etc., taking an examination, chiefly to prevent cheating; proctor.
a person who admonishes, especially with reference to conduct.
This doesn't tend to happen in most schools in English speaking countries, so be aware of the cultural differences.
There is the word "prefect". The actual role varies from school to school and was different in past times. In the past, prefects were responsible for class behaviour. In some schools there were even allowed to beat other pupils with a cane. Fortunately this doesn't happen any more, and prefects are more likely to be leaders of charity fundraising and sit in a school council, to represent the pupil voice.
Nevertheless, in schools where a student is given responsiblity for other pupils behaviour, the term "prefect" could be used to describe them.