A common relationship might be that of dominance by the mistress. She gives orders and the slave has to follow. But in some stories, the slave could have unique skills or knowledge that the mistress values. This could lead to a relationship where the mistress respects the slave to some extent. Also, there could be a relationship where the mistress feels pity for the slave and tries to make his life a bit easier, without completely breaking the social norms of the master - slave relationship.
The relationship could start as a typical master - slave one. But perhaps the slave has a kind heart and is always helpful to the mistress, which softens her attitude. Maybe the mistress then begins to protect the slave from other cruel masters or villagers. They might share stories and secrets, and the slave could become a confidant of the mistress. This would transform their relationship into a more emotional and less one - sided power dynamic.