Who is typically in charge of a typical US graduate school?
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.