I'm confused about the college application process?

If I am a high school student and want to go to a university because they excel in a certain department do I apply for the university and then go to its department automatically based on my major choice or is it something else?
For example, if I want to go to study business at NYU do I attend classes at Stern or…?

At NYU you would apply to NYU and indicate Stern as your desired school. All business classes are offered by Stern but you also take some arts and science courses.

It depends on the university so you’d have to look up the school’s policy or call them. For some, you just apply to the university as a whole and then you take classes for your major at the certain department. For these schools, it’s usually easier to switch majors to a completely different major. For some other schools, you need to apply to particular college/department within the university. You may have to apply for another college in that university if you wanted to switch majors midway.

I know Carnegie Mellon does this: let’s say you applied to the business college as your first choice and to the computer science college as your second choice. You can get accepted/rejected/waitlisted to one college and accepted/rejected/waitlisted to the other. They mainly do their admissions independently of one another.