A Board of Education is the city or prefectural body responsible for the management of schools of their particular area.
Some ALTs are based at a single school, while others will work from their Board of Education. In the simplest of terms, the difference between being based at a Board of Education and a school lies in where one goes to work on days where there is no school. An ALT based at a school will spend school vacations at their school, whereas an ALT based at a Board of Education will report to the their office usually located in their town's City Hall.
A BOE JET's supervisor will most likely be somone who works at the BOE as opposed to a JTE. A BOE JET by definition will most likely be teaching at several schools, and will be required to come back to the BOE on a certain afternoon of the week.