It is possible that both the child (or more technically, her parents) and the school may be liable.
First, the child. The mere fact that person A injures person B does not in and of itself make A liable (legally responsible for) B’s injury. Instead, there must be fault. Fault in a case like this is usually in the form of negligence, or unreasonable carelessness, or an intentional or deliberate act. The latter situation is often straightforward; for example if A punched B, A will be liable for assault—for deliberately hurting another. However, negligence is less straightforward, especially when children are involved, since what is reasonable is judged by the standard of a child of that age. So, for example, say that during a field hockey game, A accidentally knocked B’s teeth out with a wild backswing. An adult might be held to be negligent, for swinging wildly or not making sure the area was clear; but it could easily be the case that an 8-year-old girl might NOT be considered negligent, because by the standards of a child playing field hockey, she did nothing wrong.
Assuming that either a deliberate act or negligence can be shown, the child may be liable. Since parents are liable for the acts of their minor children, you may be able to sue the family.
Second, the school. Schools have a duty to properly supervise children. When they don’t, they may be liable for “negligent supervision.” However, the facts or circumstances will determine whether the school did anything wrong. For example, say that child A has been bullying and threatening child B; in that case, it would be negligent for the school to not take steps to keep A in check and prevent her from hurting B. The school is negligent because they’ve had warning that something may happen. On the other hand, say that child A just found out her parents are divorcing and, without any warning or prior incidents, “lost it” when B inadvertently made a cutting remark about divorce. Even if A then deliberately assaulted B, the school had no reason to be on its guard, and so the school would almost certainly not be liable.
If your child was injured by another, you should consult with an attorney, to see what recovery or recourse may be available to you.