The 'Authority' Project is a mysterious project which involved taking children that were 13 years old or younger to an unspecified location for unknown reasons.
At one point, three people burned documents that the occupying German forces needed for the 'Authority' project, an act that led to the people of Wellington Wells summarily executing three townspeople accused of doing so to appease the Germans.
Percival Hastings was too old to go, but he ended up volunteering to accompany Arthur. Arthur, however, didn't go by impersonating Percy and selling a lie to Constable Peters, who was suspicious but went along with it because he didn't want to send one extra child. Percy was ultimately abandoned to the Germans.
Not many people knew what the Authority Project was about, and even one of the main enforcers of the project, General Robert Byng, admitted that he did not know the true meaning of the Project. According to him, only Colonel Von Stauffenberg knew the truth about the Authority.
Mrs. Boyle poisoned her family to keep her daughters Elizabeth and Anne from being taken by The Authority.
It's never explicitly stated in the game what happens to the children. It's more or less left up to one's own interpretation, and it's unknown as to whether or not it's going to be covered in any future DLC's.
The people of Wellington Wells were deeply traumatized by this event, to the point where they couldn't have children without being reminded of what happened to the children who were sent to Germany.