What is needed to change the custody of a child the mother is unfit and the father who was never married to the mother takes very good care of the child. The child is 10 years old and wants to move?

Unless you are the mother or father of this child, the easiest way would be to make a report with Child Protective Services, who would evaluate the mother's home, if your accusations are founded, and can make a recommendation for a change in custody to the court.