Beginner is when you have just entered the academy. You stay at this level until told otherwise by myself.

Intermediate is when you have been promoted as a student. You are taught at a higher level than beginners.

Honor is when you are to graduate at the end of the year and be assigned to the Gotei 13 alone, as we cannot sort out the other two properly (Lack of details in both anime and manga).
You are taught at the highest levels the Academy has to offer.