This information sheet concerns two separate documents: one is "visa" the other is "residency permit".
"Visa"="Schengen visa"=entry permit (to Hungary and the Schengen region) -- some of you must, others are expected but not required while few of you do not need to get.
"Residency permit" = a permit to stay long term (more than 90 days) in Hungary -- all of you must get it.
Both documents are issued by the Hungarian Immigration Office.
Hungary is one of the Schengen area countries which handle jointly their immigration policy and procedure. The set of Schengen countries does not equal to the set of countries of the European union: there is a non-empty intersection but neither contains the other. For a full list of Schengen countries see https://en.wikipedia.org/wiki/Schengen_Area#Current.
The borders between Schengen countries are fully open, just like they are within the US when crossing from Ohio to Michigan, for example. Therefore once you enter any of these countries, you are free to travel within the Schengen region and any of these countries are entitled to issue so-called Schengen visas (entry permits). This is exactly why getting a Schengen visa is so much controlled and therefore difficult (just like it may involve a lot of paperwork for non-US citizens to enter the US).
All students must obtain a residency permit in order to study at BSM in the Fall and Spring semesters (since you will be staying in Hungary more than 90 days). You will get the residency permit already in Hungary.
However, most students need to have their residency permit pre-approved and obtain a so called D type visa, while they are still outside of Hungary and the Schengen zone.
Wether you need to have your Hungarian residency permit pre-approved and obtain a D type visa depends on your citizenship.
Below you will find three sections explaining the various procedures.