What power do student councils typically hold? And what role do they have?
Austria is in a rather interesting situation regarding this. Traditionally, Austrian universities were mandated to maintain what was called (roughly translated) "university democracy", meaning that all members of the university (students, non-faculty scientific employees, non-scientific employees, faculty) need to be able strongly influence all decisions that directly impact them. In practice, this led to (for instance) student representatives with quite real powers, such as vetoing certain decisions.

After 2002, a new legisation was put in place to "americanize" the system, which (interestingly) was also understood to mean that decision power should be centralized more or less purely in the faculty (in Austria that's basically only full professors). Since then, the role of student representatives has shrunk perpetually. They are still in most meetings, but if they disagree, the only consequence is a note in the meeting protocol. This has certainly led to some developments that were quite frankly not in the student's best interests.

That being said, in Austria, student representatives are still your best bet when you have some "systemic" problem with a lecture or a professor. Not necessarily because they have any power to change things, but at least they are on first-name basis with the people that do (the dean of studies, the responsible department head, etc.) due to sitting on many meetings with them. Even if they can't force anything, they should know who to talk to, who might be able to remedy the situation.