Who is typically in charge of a typical US graduate school? I was studying the website of the Harvard Graduate School of Arts and Sciences ...

Obviously an impressive institution with a broad scope. But I wonder who is in charge of the operation. For example, on the ABOUT page, I see that there is an interim Dean of the GSAS. Does this individual manage at the highest level or is there someone between her and the professors, e.g. a  kind of middle management (for lack of a better term)?

Id like to learn how US universities structure their institutions, scientifically and bureaucratically. Obviously there are differences between universities, but I’m curious what you know about your university or a university with which you are familiar
There is no single governing board for all US universities. Every university is different. 
In very broad strokes, as your question is quite broad here's how it breaks down.
Board: The Board of Trustees or University Board functions in much the same way a company board does. They meet regularly and make large decisions on spending, hiring, etc.
Chancellor: In some schools, this is the President. Think of them as the CEO that advises the Board and executes the vision.
Provost: Usually the ultimate head of academics in a university.
Deans: Usually lead various colleges. For instance a Dean of the Law School or Dean of Undergraduate Admissions

There are various various administrative staff within all of these. And again, all schools vary in terms of nomenclature.