Any hospital, college or university that teaches nursing is referred to in a generic sense as a nursing school. In order for any hospital, college or university school of nursing to teach nursing so that the students who they graduate can take the licensing exams to become an RN (or LPN), they must meet certain specific requirements of the state board of nursing for the state in which they reside. Once the school of nursing meets the state board of nursing requirements, then the individual schools are free to supplement their nursing programs of study with whatever other courses and subjects of study they decide they want in order to confer a degree of associate, bachelor, master, doctorate or PhD. Some universities are known for the unique qualities of their particular nursing curriculums, so it is something you might want to explore when looking for a nursing school.
You can become an RN by going to a community college and earning an associate degree in nursing (ADN) or going to a university and earning a bachelor's degree in nursing (BSN). The difference will be in the number of classes and credit hours each school will require you to take to get those degrees. Bachelor degrees require more classes and credit hours than associate degrees. The basic preparation to become a nurse, however, will be the same and be taught in both types of schools. Having done both, I can tell you that the difference was that in the BSN program I took, we had a much more intensive concentration of classes in psychology, communication, leadership and management than I had when I earned my ADN. We did an awful lot of writing and researching of papers for the BSN program as well. This "bachelor degree" work is something that is required by each particular college or university--not by the state boards of nursing. So, if you are thinking of getting a BSN, you are doing it to not only become a nurse, but to get a bachelor's degree along with the nursing as well. Without a doubt, most RNs are trained in associate degree programs.