(Why am I always the one to answer this sort of question? Oh, yeah, that’s right - because I attended one. lol.)
A “minor seminary” is distinct from the “major seminary” where one engages in more advanced philosophical and, particularly, theological studies. Typically, they are considered “preparatory” seminaries which enable a young man or boy to engage in personal and spiritual formation appropriate to his age and experience that will prepare him for eventual enrollment into the major seminary program.
A “minor seminary” can take a few forms.
In Chicago, for instance, we have ethnic houses of formation for Spanish speaking and Polish speaking men (as well as a couple of other groups) where they can better inculturate and learn English well enough to succeed in future studies, while going through human/spiritual formation. These preparatory institutions which are part of the seminary system are essentially “minor” seminaries.
Sometimes today college level seminaries are even considered minor seminaries, though actually they are typically institutions more akin to the philosophical study that often used to take place (and still does in some institutions) at the major seminary.
More typically (and historically), however, “minor seminary” refers to a high school level instituion which educates and forms boys in preparation towards the priesthood and what will be required of them at the more advanced levels of seminary schooling.
Quigley Preparatory Seminary (“Le Petit Seminaire” as it was lovingly known), in Chicago, was one example of such an institution - which lasted over 100 years; until just a month ago. As the carving in stone above the entrance arch still reads: “Built for the Glory of God and for the education of youths for the priesthood.”
Nowadays, however, such minor seminaries are, sadly, few and far between - despite the Church’s encourangement and endorsement of their importance and necessity. Today, what was the nation’s largest and most notable no longer even exists.
quigley.org
en.wikipedia.org/wiki/Archbishop_Quigley_Preparatory_Seminary