For a single incident of assault, the agressor may be taken to court both through the criminal justice system (criminal law) and through the public justice system (civil law, in the branch of torts).
Criminal law and civil law differes in many ways, but the main point is that in criminal law, the police are the people who initiate the case, and are prosecuting a member of society (or a member of society for thepurpose of the case). In a civil case, there is a member of society raising an issue between another member of society and seeing to apply legal principles to achieve a mutally agreeable and just conclusion, and is a case in which there is a plaintiff (or a plaintiff party) and a defendant (or a defending party).