Injury, hurt, damage, harm, mischief mean the act or the result of inflicting on a person or thing something that causes loss or pain.
Injury is the comprehensive term referable to an act or to a result of that act which involves a violation of a right or of health, freedom, and soundness of body or mind, or causes a partial or entire loss of something of value.
Hurt, applies basically to a physical injury (as a wound, lesion, or contusion) that results from a hit, a stab, or a blow. In extended use hurt applies chiefly to an act or result that involves pain, suffering, or loss; thus, a person whose rights as an heir have been violated may be said to suffer an injury but not a hurt; a person whose reputation has been damaged by a false rumor has suffered both an injury to his business and a hurt to his feelings; a dentist in drilling a tooth may cause a hurt, but not commonly an injury.
Damage applies to an injury that involves loss (as in property, in value, or in usefulness).
Harm (usually without an article) is referable to an evil that injures or may injure; often it suggests a consequent suffering (as grief or shame).
Mischief carries a stronger reference to the person or thing that works harm or is capable of inflicting injury; it applies either to the harm or injury that results from an agent or agency or occasionally to the aspect of a situation that causes harm or vexation.