The answer is different depending on a number of factors. First, it depends on whether the child is being prosecuted for breaking a criminal law or sued for something civilly. In the case of criminal activities, most crimes have an element requiring intent. Since a parent is not an actor in the crime and, therefore, lacked the intent for the crime to occur, they cannot be prosecuted for the crime. In a few instances, however, if the crime only occurred because of the parent's lack of care for the child, the parent could face some sort of charges, albeit, for a different crime like child neglect.
The child, of course, will also face different types of charges and penalties based on his or her own circumstances, like age, demonstration of knowing right from wrong, prior record, and so forth. In some instances, a child may not be charged at all if they were so young that the incident was a clear mistake, such as a young child who gets a parent's gun and shoots someone. If a child is old enough to know better, they will normally be charged as a child and face juvenile sentencing. These penalties are usually lighter than for adults and, if they involve incarceration, the child is typically sent to a facility for juveniles. If, however, the child showed a clear understanding that what they were doing was wrong, particularly in more serious crimes like murder, the child may be tried and sentenced as an adult.
In civil cases, on the other hand, the situation is usually reversed. Children are rarely responsible civilly for their bad actions, but rather, their parents end up having to foot the bill. For example, if a 12-year-old gets into a fist fight in the school yard and knocks another child's tooth loose, the injured child could sue the young ruffian's parents for the injury. From a policy standpoint, this makes a great deal of sense. After all, a 12-year-old is not likely to have any meaningful source of income, and would likely be unable to pay for the medical expenses. Since the point of civil liability is to try to put the injured party in the position they would have occupied had the injury not occurred (or as close thereto as possible through the payment of money), it would make little sense to limit the scope of liability to a child who cannot make restitution. Indeed, from a legal analysis standpoint, it is similar to the liability of a pet owner for injuries their animals may cause. In both instances, the actor actually causing the injury lacks legal capacity to choose to cause the injury and has no means of providing a remedy for the bad action. Thus, it falls to the one responsible for training the child or pet about the appropriate means of behaving in public to pay for the harm caused by the lack of discipline.
