In the United States, the National Transportation Safety Board establishes the criteria, and they're spelled out in Title 49 of the Code of Federal Regulation.
Aircraft Accident is defined by 49 CFR 830.2 as:
"Aircraft accident" means an occurrence associated with the operation of an aircraft which takes place between the time any person boards the aircraft with the intention of flight and all such persons have disembarked, and in which any person suffers death or serious injury, or in which the aircraft receives substantial damage. For purposes of this part, the definition of "aircraft accident" includes "unmanned aircraft accident"
Aircraft Incident is defined as:
"Incident" means an occurrence other than an accident, associated with the operation of an aircraft, which affects or could affect the safety of operations.
A fatal injury is any injury resulting in death within 30 days of the accident.
Serious Injury is defined as:
"Serious injury" means any injury which: (1) requires hospitalization for more than 48 hours, commencing within 7 days from the date the injury was received; (2) results in a fracture of any bone (except simple fractures of fingers, toes, or nose); (3) causes severe hemorrhages, nerve, muscle, or tendon damage; (4) involves any internal organ; or (5) involves second or third degree burns, or any burns affecting more than 5 percent of the body surface.
Substantial Damage is defined as:
"Substantial damage" means damage or failure which adversely affects the structural strength, performance, or flight characteristics of the aircraft, and which would normally require major repair or replacement of the affected component. Engine failure or damage limited to an engine if only one engine fails or is damaged, bent fairings or cowling, dented skin, small punctured holes in the skin or fabric, ground damage to rotor or propeller blades, and damage to landing gear, wheels, tires, flaps, engine accessories, brakes, or wingtips are not considered "substantial damage" for the purpose of this part.
Accident criteria for unmanned aircraft are also spelled out.
The NTSB provides further definition and requires immediate notification in the event of any of the following items (49 CFR 830.5):
§ 830.5 Immediate notification.
The operator of any civil aircraft, or any public aircraft not operated by the Armed Forces or an intelligence agency of the United States, or any foreign aircraft shall immediately, and by the most expeditious means available, notify the nearest National Transportation Safety Board (NTSB) office1 when:
(a) An aircraft accident or any of the following listed serious incidents occur:
(1) Flight control system malfunction or failure;
(2) Inability of any required flight crewmember to perform normal flight duties as a result of injury or illness;
(3) Failure of any internal turbine engine component that results in the escape of debris other than out the exhaust path;
(4) In-flight fire;
(5) Aircraft collision in flight;
(6) Damage to property, other than the aircraft, estimated to exceed $25,000 for repair (including materials and labor) or fair market value in the event of total loss, whichever is less.
(7) For large multiengine aircraft (more than 12,500 pounds maximum takeoff weight);
(i) In-flight failure of electrical systems which requires the sustained use of an emergency bus powered by a backup source such as a battery, auxiliary power unit, or air driven generator to retain flight control or essential instruments;
(ii) In-flight failure of hydraulic systems that results in sustained reliance on the sole remaining hydraulic or mechanical system for movement of flight control surfaces;
(iii) Sustained loss of the power or thrust produced by two or more engines; and
(iv) An evacuation of an aircraft in which an emergency egress system is utilized.
(8) Release of all or a portion of a propeller blade from an aircraft, excluding release caused solely by ground contact;
(9) A complete loss of information, excluding flickering, from more than 50 percent of an aircraft's cockpit displays known as:
(i) Electronic Flight Instrument System (EFIS) displays;
(ii) Engine Indication and Crew Alerting System (EICAS) displays;
(iii) Electronic Centralized Aircraft Monitor (ECAM) displays; or
(iv) Other displays of this type, which generally include a primary flight display (PFD), primary navigation display (PND), and other integrated displays;
(10) Airborne Collision and Avoidance System (ACAS) resolution advisories issued either:
(i) When an aircraft is being operated on an instrument flight rules flight plan and compliance with the advisory is necessary to avert a substantial risk of collision between two or more aircraft; or
(ii) To an aircraft operating in class A airspace.
(11) Damage to helicopter tail or main rotor blades, including ground damage, that requires major repair or replacement of the blade(s);
(12) Any event in which an operator, when operating an airplane as an air carrier at a public-use airport on land:
(i) Lands or departs on a taxiway, incorrect runway, or other area not designed as a runway; or
(ii) Experiences a runway incursion that requires the operator or the crew of another aircraft or vehicle to take immediate corrective action to avoid a collision.
(b) An aircraft is overdue and is believed to have been involved in an accident.