What is a "graduate school"?
In US universities, a 'graduate school' is usually an administrative entity that manages all aspects of graduate life. At the University of Utah, for example:


  The Graduate School fosters excellence by providing administrative
  structure and leadership to maintain and enhance graduate education at
  the University of Utah.
  
  Our programs offer financial assistance, rigorous academic
  opportunities, and professional development to students, staff, and
  faculty.
  
  We are guided by the principles of quality, diversity, and integrity
  as we help students to prepare themselves for successful, relevant
  careers.


The link posted in the comments suggests that faculty are required to be associated with the grad school in order to function as advisors etc. I think this is also an administrative action, designed to demarcate faculty that can advise students from all the other people who are designated as faculty (for example, adjuncts, research faculty, teaching faculty, and so on). 

So there's no academic function associated with the graduate school. 

You can think of the graduate school as the university-wide entity that manages the individual chairs of graduate studies at departments.