Board of Visitors
A Board of Visitors is a body of alumni donors who serve their respective schools in an advisory capacity. For example, the Columbia College Board of Visitors is 40+ strong and is supposed to "provide alumni support for the goals and activities of Columbia College... [and] seeks to support and counsel the President of Columbia University, the Dean, Administration and Faculty of the College and related senior staffs and play an active role in the establishment and maintenance of communication links among College students, faculty, administrators and alumni."
The term likely stems from the British Academic Visitor.
Columbia Law School is the only school where its board has governance right as mandated by the ABA. Specifically, the President of Columbia University cannot simply appoint the Dean of Columbia Law School as he may for the other schools. The Dean must be chosen and elected by its governing board. For the other schools, although boards assist in the search process, the President retains the final say.