So just out of lame curiosity...what's the difference between someone being a faculty member with the title of "Instructor of Anesthesiology" vs. "Professor of Anesthesiology"?
So it has nothing to do with whether an anesthesiologist is BE (board eligible, as in couldn't pass their boards) vs. BC (board certified)?
As others have said, it is one (the lowest) in a series of academic titles. The criteria for advancement from one to the next will vary by university. For example, one university I know of (I work there) appoints you as an Instructor if you don't have a fellowship and an Assistant Professor if you do have a fellowship (I think we even pay the Instructors less). This is arbitrary in that some other university might have a different criteria (e.g., board-certification, full/part time, etc.) for distinguishing between the two.
So it has nothing to do with whether an anesthesiologist is BE (board eligible, as in couldn't pass their boards) vs. BC (board certified)?