What happens to Jonas when he turns 13?

Jonas effectively becomes an adult at the age of thirteen. It's at that point that one's precise age becomes irrelevant in this community. As Jonas's mother tells him, age becomes unimportant once you hit thirteen; after that, you lose track of how old you are and have to access your precise age by consulting the Hall of Records.
As Jonas has now hit this unlucky age, he must continue with the career assignment first given to him when he was twelve, which he must carry out for the rest of his life. Jonas's assignment is, of course, that of Receiver of Memory. This entails acting as a repository for the community's memories so that the past can never be completely forgotten, even though the community elders have rejected so much of it.