@raifive depends on the school really. But if they have it (a "class-representative"), it usually is quite different from the jp concept. The ones I know of from here is usually instead designated ppl that students can bring up issues with, and they note it down and bring it up in some meeting between the other classes representatives and teachers. Basically it is just another feedback system like the "feedback box" (with the advantage that it is less anonymous (albeit still more than going to a teacher), and the same disadvantage), that also is more focused at bringing up issues or inquiring about questions than the box's that usually are more for suggestions.
If the school has a "student council", odds are it consists of one person from each class, making them the representative of said class (only time I had to vote for something resembling a student council, was in uni, where the student council equivalent was a split entity from the ones dealing with study-issues). Meaning the term class-rep would instead refer to the student council members.
They are not there for carrying papers or taking charge of events (ever remember all school-festival arcs in mangas?). Though it often is the same kind of ppl who decide to be one that is also the kind that suddenly decides to arrange a skiing-trip with classmates or a student-disco (and sometimes manages to convince some teachers to help, or appropriate school-infrastructure for it, despite it being a personal event).