The opposite of a nonsectarian school is a sectarian school. It will describe its religious affiliation as Roman Catholic, Baptist, Jewish and so on. Examples of sectarian schools include Kent School and Georgetown Prep which respectively are Episcopal and Roman Catholic schools.
Why is this important? Does it matter whether a school is sectarian or nonsectarian? This is information which you need to know when you choose a school. For example, let's say you are Jewish. You probably would find it difficult to have to sit through a Roman Catholic mass if you decided to go to Georgetown Preparatory School. It's been done before, but it is something to consider seriously. There would have to be lots of good reasons besides religious affiliation why Georgetown was the best fit for you.
Find more terms in the Private School Glossary.