A Hall Director is a student (usually a grad student) employed by the University Department of Residential Life (aka "Housing") to be ultimately responsible for the safety and well-being of a Residence Hall (aka "dormitory"), its student residents, and a staff of Resident Assistants.
If a Director is doing his or her job properly, you should see him out and about, talking with students on a regular basis. A good 50% of the Hall Director position is social: knowing each resident by name, knowing what issues are cropping up between Roommate A and Roommate B, taking the time to wish someone a happy birthday, keeping the boisterous students somewhat in check while making sure the quiet students get properly integrated into their new environment.
Hall Directors are usually compensated for their work with a stipend and/or tuition waiver. They are generally required to live within the dorm they manage so they can be available at a moment's notice should something go wrong or should they be needed in any way. Modern dorms are built with small, studio apartments (usually on the first floor) to accommodate this living requirement.
While a Hall Director's main purpose might be to protect the dorm and its inhabitants from any fires, illegal activities, potato gun incidents gone horribly awry, etc., s/he is also responsible for the management of a team of Resident Assistants, or RAs. These Resident Assistants live with the residents (often receiving a free private room in return for their work with Res Life), perform Health and Safety inspections on a regular basis, provide social and educational programming (parties, study sessions, etc.), and primarily "keep an eye" on how everyone in the dorm is doing.
Depending on university rules and regulations, RAs may take care of calling in repair requests and/or inspecting rooms for contraband materials, but these higher-level responsibilities are usually reserved for the Hall Director. In most dorms, the Director is also responsible for holding meetings with the staff, presiding over Hall Council (a governmental body of elected dorm residents who plan and execute various activities), training new staff, and providing detailed reports to his or her immediate supervisor (often called an "Area Coordinator").
I was a Hall Director from 2000 to 2002 while I was completing my MBA at the University of Alabama. Although we had our fair share of late-night fire alarms, fights, drug mishaps, roommate conflicts, and the like, I treasure the time I spent at UA's Friedman Hall. If anyone reading this is considering becoming a Hall Director (or a Resident Assistant), I'd highly recommend it. It's a tremendous (and, dare I say, life-altering) experience, and you're guaranteed to have a bunch of great stories to tell your friends for years to come. (Plus, the experience looks great on your resume or CV.)