A restitution hearing is a criminal hearing in which the court determines by the evidence provided what amount of restitution the defendant must pay to the victim. The amount can come straight out of pocket or it can come from insurance proceeds or other sources. If there is an identifiable victim there will more than likely be a restitution hearing. Restitution hearings will probably not occur if there is no damage to the public, for example you are charged with drunk in public and no damage to property or drunk driving and no accident occurred.
Restitution hearings occur after conviction or a plea and can occur a number of months later. The victim is allowed to speak/be present at the hearing and can present evidence as to the amount of damages they suffered. The court can accept or reject this evidence depending on credibility and relevance.