What exactly happens when a department head is contacted about a specific instructor in a specific course?
Expanding on my comment to make it an answer.

Department chairs deal with this all the time, and have to figure out which complaints are legit and which are students complaining as an excuse. I've seen several courses of action.


Do nothing. If the student's complaints don't seem reasonable or seem out of character for that instructor, it's a waste of time to worry about it.
Meet with the student to find out more information. I've seen this happen when a complaint is delivered by email, but at my university it's rare because most student feedback happens via anonymous surveys at the end of the semester.
If it's a one-time event, talk with the instructor involved. Students provide one side of the story; an instructor might provide a different side. I had this happen when a student complained about my grading of an assignment -- turns out they had received a mandatory 0 for plagiarism and were hoping a complaint would get them a better grade. If the complaint is that the material or exams are too hard, it would be appropriate for the chair to review the material (perhaps with a professor familiar with the subject). For example, my courses have a course co-ordinator who could impartially review any questions about my lectures, assignments, tests, or grading.
If it's a pattern of legitimate complaints, the chair can take either mentoring or disciplinary actions as allowed by the university. My university has a faculty mentoring program where other faculty members will sit in on classes and offer help with improving teaching styles. Non-tenured faculty can also be progressively disciplined in a number of ways -- tenured faculty are obviously harder to discipline. Ideally someone doesn't get tenure if they have obvious problems in the classroom, but we all know it does happen. I've seen department chairs "punish" tenured faculty by giving them less desirable teaching assignments, but it also doesn't seem really effective.


Ultimately, it's going to be a combination of how credible the complaint is, how many complaints there have been, how motivated the chair is, and what options are legally available to the chair.