It all depends on where you choose to work. If you work in an outpatient setting, you would see kids with minor illnesses and injuries (e.g. colds, flu, etc.) ... kids needing physical exams for schools, camps, etc. ... kids needing immunizations ... kids need following, monitoring, management, of chronic illnesses such as diabetes, GI problems, etc.
If you work in a children's hospital, you could see anything -- depending on the unit you work on. You would find kids with cancer on the oncology unit, etc. There are units that specialize in the care of every kind of patient with every kind of health problem -- kidney disease, heart defects and heart disease, infections, respiratory illnessess such as asthma, GI problems, kids having transplants, etc. There would be kids having surgery, kids requiring intensive care, a unit specializing in the care of newborns, etc. You could see just about anything in a major children's hospital.
What types of patients you see (and their diagnoses) depends on where you choose to work.
What llg has said is true, but... In general terms, most children who require nursing care will have a respiratory issue due to their small airways and immature immune systems. Second in line would be dehydration (nausea, vomiting and diarrhea as contributing factors). Third is seizure, whether febrile, traumatic or metabolic. Fourth is traumatic injury (including burns) and fifth is failure to thrive. Each of these may be related to some other systemic problem, such as prematurity, congenital anomaly, very young and unprepared parents or having multiple siblings. But the presenting problem - the reason Mom or Dad has sought health care - is typically one of these.
In my practice I see mainly cardiac patients, but having three grown children, one of whom being medically-complex, I've spent LOTS of time in children's hospitals.