For more info on the hours aspect of residency, check out the "Hopkins IM loses accreditation" thread. Most hospitals seem to be honoring the 80 hour a week regulation, and I imagine, after the Hopkins situation, programs are feeling more and more pressure to do so.
Residents typically work as a team; for example, a team might have one senior resident and two interns, and, in their care (for which the interns have most of the day-to-day responsibility) they may have up to 20-25 patients (on something like internal medicine). Surgery teams are typically larger, with, maybe, two interns, a senior resident, and a chief, with the more senior residents providing the direction and interns performing the majority of the day-to-day duties.
Residencies are required to provide five hours per week of didactic instruction (lectures, etc). Residents, in turn, may present at these conferences, and also have teaching responsibilities with medical students rotating on the service (i.e. working with the team), particularly the more senior residents.
Most residencies give yearly exams in preparation for the boards examinations, which resident take at the completion of the residency. And, while many residencies are three years in length, this is the minimum amount of time required in a residency; some may be longer, and, when considering fellowships, may extend for a very significant amount of time (an internal medicine residency plus a cardiology fellowship reaches at least seven years).
Residents' responsibilites increase as they progress through the residency. Residents provide backup for each other and double-check each other's work. If you're working with the right people, the team concept works very well, and things tend to go smoothly. Interns (first-year residents) typically see a new patient first, do the examination and write-up, write the patient's admission orders, and get everything going. The supervising resident then "fine tunes" things and provides general direction for the patient's care. The attending physician (the most senior doctor-- non-resident) gives the final word; what he or she says goes. The attending also carries the majority of the malpractice liability for the patient's care, so if the attending says something should happen (i.e. a test, a treatment, etc.), it happens.
Hope that's helpful.