My interview was a very very long time ago, so much of it is no longer relevant in 2019 [e.g. I got my job through advertisement in the New York Times!]
But, some parts of the process are still standard, plus about 10 years ago I had to interview internally for a different position. The latter is primarily what I include here.
Ultimately, a lot of the interview & outcome depends on the final interview with the person you would be working for, e.g. whether or not they are a good interviewer.
The relevant staff person/team usually develops the job description, with input from HR. HR screens applications/resumes & provides selected applications to the staff/team.
Sometimes the next step is an interview with an administrator/executive from the overarching program or department, or, as relevant, other staff from the team.
For example, as an admin assistant, I was first interviewed by the Executive Secretary of the VP of the department I'd be working in. In this case, some of that interview covered skills aspects like relevant experience & typing/spelling testing, and some I think was to check out personality & 'fit' for the organization.
For a higher level position, I'm sure one would be interviewed by several relevant staff from the team/department, either individually or as a group.
My interviews were all on the same day: first with HR person, then high-level administrative staff from the department, then with the Associate for whom I'd work. In that interview, though she did need to confirm my technical/computer skills - as there would be a lot of preparation of documents/manuscripts, presentations, & transcription - she was less interested in reviewing my skills & more about other aspects of my background that would show my interest & fit for the position (e.g. relevant volunteer work, activism, personal interests, etc.).
Unfortunately I can't remember so many of the interview questions.