You are responsible for obtaining your study visa for Florence. Keep in mind that the Italian Consular Service offices are not affiliated with New York University. They are entities of the Italian national government that issue visas and perform other official functions, and they may create and/or change their requirements for obtaining a student visa at any time and without prior notification. Therefore, the steps below are meant as a general guide but it is your responsibility to follow the most up-to-date and accurate instructions as reflected on the Italian Consular Services official web sites.
What is a student visa?
It is an official stamp issued by the Italian government that gets affixed to a page inside your passport before you depart the United States. You must present it to immigration control when you arrive in Italy.
You must have a study visa because it is not legal to study as a full-time student in Florence without it.