Forum admins are often the forum owners. So they make money using ads or membership. I have an article on how forum owners can make money.
As a moderator, one may be hired by the administrators for moderating the forum. To become a moderator, one may need to contribute to the community before an admin or forum owner wants to hire a member to become a moderator.
Also, forum owners may hire forum admins to overlook the day to day activity in the forum.