Non-binary is one of many terms used to describe people whose gender identity is neither entirely male nor entirely female. Individuals who are non-binary may use other terms too like gender non-conforming, queer, or genderqueer.
Many non-binary people don’t use he or she pronouns because they don’t feel right. Some people use they, for example. Other people may use he or she but still not feel internally that they are entirely male or female.
For more information on transgender people and the trans movement, check out the National Center for Transgender Equality.