The Church or Conclave is an organization within the world that takes it upon itself to observe, preserve, or exterminate the supernatural entities that encompass it for the sake of humans. It is a multi-faceted group divided into factions.
Established far into the past, the Church was took it upon themselves to deal with anything that endangered humanity, such as Devils and Demons and Faeries and the like. They had a very aggressive stance during this time period which brought them in contact with Elias, who noted that they were far less strict in the present and attempted to harshly interfere with his life.
However, in the present they've taken in multiple religions, even those native to the Far East, and would rather stick with either observing them or preserving a cordial relationship if possible. That being said, there is a lot of politicking and the peaceful means they are currently using aren't to every faction's liking, resulting in a lot of power plays.
The Church itself is divided into three sections based on the three roles they take: Observation, Preservation, and Extermination. Gabriella refers to them as essentially a collection of private armies.
The observation segment consists of agents known as pigeons or roosters, who are scattered all over the world. They observe the supernatural entities and those who work with them to see if they are a threat who harms humans.
The preservation segment consists of agents known as crows or ants, who work to make contact with and preserve cordial relations with the more dangerous the supernatural entities, reaching a compromise that benefits both parties.
The extermination segment consists of agents known as wolves or locusts, who are called in to remove the supernatural entity if a compromise cannot be reached and protect the people. Members of it likely contain abilities suited to bind or exterminate non-humans.
The Shepards are the ones who stand at the top of the structure, the leaders. Each faction answers to a different Shepard.