Expulsion in Imperial Warfare is a lot like the kind of expulsion you may have heard of in grade or high school; it consists of kicking an enemy Priest out of your territory so that he hopefully never comes back. Expulsion is generally incited by enemy Defamation. Why else would you want to smear the reputation of a peaceful passing Priest so badly that he never shows his pious face above your Plots again?
Expulsion is simple: Simply click an enemy Priest who's trampling all over the nice green lawns of your kingdom and select "Expel." Then pick a Priest from the list of available (idle) units (preferably the one with the highest Pious attribute value?) and get that sharp tongue moving.