A Nachhilfezirkel is an independent institute, in which kids with school problems receive extra tuition. The teaching is done by retired school teachers and college students. Kids are usually taught in small groups. A Nachhilfezirkel is usually less expensive than a private tutor.
An Internat is a privately run boarding school. It is not associated with 'normal' schools and can put emphasis on subjects that may not be taught in a different kind of school. Internate are inspected by the German school board, however, to ensure that kids reach university standards. German Internate are popular for parents with "problem kids". A lot of them are run by the Church. Generally speaking, they are all expensive.