Kid Fic refers to stories where the main characters have children. This is different from fic featuring Wee!chesters - i.e. stories focussing on Sam and Dean's childhood. (see also Wee!cest)
Sam and Dean, or Jared and Jensen are generally the parents, although other Slash and Het pairings may feature. The children may be the result of male pregnancy - Mpreg. In Sam and Dean stories often children they have encountered in their hunting such as Michael, Ben Braeden or Jesse end up in their care.
RPF stories often feature Jared or Jensen as single parents, who have a child as a result of a previous Het relationship, although Mpreg stories also exist.