Villager's Fork is an item that allows characters to become "Villager" class (Some characters cannot be Villager even Villager's Fork is used).
Villager's Fork can be obtained from "Gift from Goddess Mila", one of the free DLC. According to the Nintendo's Fire Emblem Echoes official website, players can just get one Villager's Fork this time.
Get a character to have Villager's Fork and visit a Mila Statue just like when you change character classes.
For example, Forsyth the Soldier basically promotes to Knight. If you however get him to have Villager's Fork, Villager is added to the classes lineup.