What does the FAA consider to be "good moral character"?
An examiner uses the contents of FAA Order 8900.1 - Flight Standards Information Management System (FSIMS)as guidance for doing their job.

As you might imagine, there is some guidance available for them there:


  
  VOLUME 5 (AIRMAN CERTIFICATION)
  CHAPTER 2 (TITLE 14 CFR PART 61 CERTIFICATION OF PILOTS AND FLIGHT INSTRUCTORS)
  Section 18 (Conduct an Airline Transport Pilot Certification, Including Additional Category/Class Rating)
  Paragraph 5-704 (ELIGIBILITY –ATP CERTIFICATE – AIRPLANE, ROTORCRAFT, AND POWERED LIFT):
  C. Good Moral Character Requirement:
  An applicant must be of good moral character. The inspector must ask an applicant if the
  applicant has been convicted of a felony. If the applicant’s answer is
  affirmative, the inspector should make further inquiry about the
  nature and disposition of the conviction. If an inspector has reason
  to believe an applicant does not qualify for an ATP certificate
  because of questionable moral character, the inspector must not
  conduct the practical test. Instead, the inspector will refer the
  matter to the immediate supervisor for resolution. The supervisor may
  need to consult with regional counsel for a determination concerning
  whether the applicant meets the moral character eligibility
  requirement.
  


That's the basic guidance that the examiners have, but there are also legal cases that have delved into this area and give more specific examples.  I found a blog which covers it in some detail at All About Airplanes.  Falsification of documents, embezzlement, and acting in a malicious manner towards others are cited as reasons that ATP certificates have been denied or revoked.