The most common are local side effects including pain, swelling, and tenderness on the arm where the vaccine was given.  Fever and other side effects like headache, fatigue and nausea can occur but are less common in children than adults. Less than 7% of children had a fever. Side effects were more common after the second dose.
The CDC Vsafe and VAERS programs closely monitor reported side effects in children and adults.  There are several other studies being conducted in the United States that are looking at side effects. All of the available data indicate that the benefits of vaccination greatly outweigh the risks, including the risks associated with children having COVID-19.