From my experience, some of the most common questions are listed below. In Commonwealth countries [e.g., Australia, UK] these are usually asked by a formal interview panel. In North America they are usually asked in informal meetings with individuals. I don't know how it works in Europe.
Why are you interested in this position?
What do you think is your most significant research achievement?
What are your scientific goals for the next 5 years? 10 years?
How will you obtain funding for your research?
Who do you think you might collaborate with at this university?
What is your philosophy of teaching?
What courses would you like to teach here?
What is your philosophy of supervision of postgraduate research students?
How do you think you could be involved in university service and community outreach?
Given the common occurrence of these questions I suggest you write out your answers beforehand and keep the piece of paper in your pocket.
It may be helpful to think of your three main selling points and try to integrate them into your answers.
There is a longer list of good questions here.
What questions should you ask?
Ones that show you know something about the department and institution, that you are interested in coming there, and that you want to be successful together.
At this stage, your goal is NOT to gather information to help you decide whether or not you would accept an offer. Your goal is to get them to make you an offer. Later you can ask hard and demanding questions about salary, start up funds, teaching loads, departmental and institutional politics, .....
Here I disagree significantly with some of the websites that show up when you Google "faculty interview questions", e.g. this one at Dartmouth College. I think the questions listed are more the type of questions I would ask if I actually got an offer. I think asking too many of these questions may irritate people and backfire. I have seen this happen.
Perhaps, my concern is a cultural difference (Australian vs. American). But I find many of those questions as too aggressive and a bit too direct for me. Some come across as rather naive and unlikely to get an honest answer. For example, if you ask a Department chair "Is this department united or divided?" I am skeptical that you will get an accurate answer.
Finally, my post-doctoral mentor John Wilkins has sage advice.
Please share your experience, either being interviewed or doing the interviewing.