Under what circumstances is an FAA accident report required?
Generally you do not notify the FAA, you notify the NTSB (the relevant regulations are in 49CFR, specifically 49CFR830).  

If you notify the FAA they will contact the NTSB, because the NTSB is the agency charged with investigating aircraft accidents.

If you crash in a cornfield somewhere and notify the NTSB they will probably ask the nearest FSDO to send an investigation team (unless you're a major airline or a fairly high-profile incident).

So ultimately it doesn't much matter which agency you call, but you should be calling the NTSB.



With the administrative stuff out of the way, the reporting requirements are also in that same section of 49CFR. Basically they say:

You are required to notify the NTSB "immediately, and by the most expeditious means available" of any aircraft accident, which is defined as:  


  …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.


You are also required to notify the NTSB (again, "immediately, and by the most expeditious means available" of any incident (something that's not an accident as defined above) which meets the following criteria:


  
  Flight control system malfunction or failure;
  Inability of any required flight crewmember to perform normal flight duties as a result of injury or illness;
  Failure of any internal turbine engine component that results in the escape of debris other than out the exhaust path;
  In-flight fire;
  Aircraft collision in flight;
  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.
  Specific requirements for aircraft with a takeoff weight above 12,500 pounds
  Release of all or a portion of a propeller blade from an aircraft, excluding release caused solely by ground contact;
  Specific requirements for "glass cockpit" displays.
  Specific requirements for TCAS advisories.
  Damage to helicopter tail or main rotor blades, including ground damage, that requires major repair or replacement of the blade(s);
  Specific requirements for air carrier blunders like landing on the wrong runway.
  


"Immediately and by the most expeditious means available" basically means pick up the telephone and call them: They don't want you dispatching a letter via Pony Express that's going to take a week to get to the NTSB's headquarters while the aircraft is sitting in a field exposed to the elements with evidence literally washing away in the rain.