Being on call while in residency means that you stay in the hospital overnight and care for the patients on your team and the other teams, and care for the new admissions. This means you will be working up to 28 hours straight with little or no sleep. This is done every second, third, or fourth night depending on the institution and specialty you are in.
Being on call for long stretches as a resident is a long-standing tradition in medicine. Many people outside the medical field see this as an unnecessary, perhaps dangerous tradition. The argument on the other side goes that the more experience you have treating patients and their diseases the better you will be as a physician. In order to see the same number of patients from 9 a.m. to 5 p.m. the length of training programs would need to be considerably longer. Also, someone needs to care for patients after hours. An old saying in medicine goes, the only thing wrong with being on call every other night is that you only get to see half the patients.
An intern is at the bottom of the food chain at a teaching hospital. Any unpleasant, menial task (called "scut" work) is the intern's job.
Another medical tradition during internship is the phenomenon of "see one, do one, teach one." In other words, when a procedure needs to be performed you see someone do it. Next time it needs to be done, you do it. The following time, now that you are an expert, you teach someone else to do it. This is how some medical knowledge gets passed on.
At this point you are finally being paid for your services. The average salary for a first-year resident in 2018 was $59,300 [source: AOA].