Where the FAA comes by to check in on you. Often happens after you have an incident. In the case of a CFI, a 709 ride can come after a student of yours screws up. One of my former co-workers had a 709 after a CFI applicant of hers landed gear up. In that case, they are looking to rule out your teaching as a cause in the accident/incident.

By the way, she did fine and now works for the FAA. 709 rides can be recovered from, especially if the reason for the ride is a less severe situation.