Rightly called a
44709 ride (taken from the CFR # and shorted to '709' as it used to be a 609) it is the FAA's authority to re-examine your competency to excercise the priviledges of one (or both) of your flight certificates. The most common 709 ride in GA aircraft is after someone does a gear up landing. My old boss had a 709 ride on his flight instructor certificate because a student he signed off to solo had a runway incursion and caused a 737 to have to go around. The FAA has extremely broad discretion (only has to show that lack of qualification/competence could have been a factor for the incident) on the reasons it can require you to submit to a 709 ride but most of the time there was an incident, maybe not an accident, and they want to check you out.
Your average 709 ride consists of generally only tasks that may have been called into question by whatever happened. The experience really depends on who is performing the 709 ride. Most of the time it is not a huge deal and the FAA just wants to make sure you aren't insane. However, if you get an inspector who for some reason doesn't like the cut of your gib you will find this to be a nightmare.
If you fail they'll generally give you one more chance after you've received dual instruction in the areas you were deficient. If you fail again or if you just failed the one time and the inspector doesn't like the cut of your gib expect to be very shortly the focus of an FAA enforcement action seeking the revocation of your certificate. (Note: This certificate would only be the one being tested for. For example, if you are a horrible teacher and you fail your 709 ride on your flight instructor certificate they will not revoke your pilot certificate necessarily.)
At that point you'd have to go through the appeals process like any FAA enforcement action. But I have heard of very few successful appeals if the reason of your revocation was the deficient performance in a 709 ride. You would almost have to show malfeasance or incompetence on the part of the FAA inspector.