The way these work is that each of the 4 shock absorber adjusters ("red things on top") are connected to a suspension ECU box in the boot (black box with Bilstein written it). The ECU is continually adjusting each shock absorber according to road speed, acceleration, braking and lateral forces (some of these from accelerometers) - 17 different adjustments according to the manual. The adjusters contain a stepper motor and some gears (there's some potential for breakage and wear/tear!), in addition to reveiving signals to make adjustments, each adjuster sends feedback to the ECU to confirm the adjustment so if the adjuster is sticking, the shock absorber itself is refusing to budge or there is an electrical connection issue then the ECU knows there is a problem, the light appears and all shocks are adjusted (or the ones that can be adjusted) to a fixed, fairly stiff setting.
When the ignition is first turned on, all 4 adjusters spin from one extreme of the adjustment to the other to confirm movement which is exactly why allandwf rightly suggested most errors appear at startup - that's when all 17 adjustment positions are used on all 4 shock absorbers. Once an error has been detected and the light appears, it stays on until the ignition is turned off and the whole process starts again with the full adjustment test once the ignition is switched on again. This is why, with an intermitting problem (e.g. slightly sticking) the light can be sometimes cleared by switching the ignition off and on again. The good news can be that even if the light does go out on a restart, the error is stored in the ECU and a good independant specialist or a dealer can interrogate the ECU via SD2 and immediately identify the offending corner even if the light is out when they get to see the car - this goes a long way to speeding up finding the issue, it's a quarter of the work immediately.
When Sport mode is selected, a different "map" is used in the ECU which leans towards stiffer settings. However, the difference when going down a straight, smooth road is prettly well undetectable, the harder you push the car around corners, the more discernable the difference becomes with significantly more grip. It can be a somewhat strange feeling at first if you are used to a car with a fixed mechanical suspension, once you get used to the "fixed" car as you corner hard, you can start to predict when oversteer will begin, with the electronic suspension the mental prediction you make becomes worthless because the ECU automatically makes adjustments changing the suspension characteristics to increase the grip - it's like "I can't believe the car managed to maintain grip there, I expected it to start to slide!" and of coursed the impressive traction control in the 3200GT compounds the whole sensation. The caveat though is that it is not recommended to use Sport mode on wet or icy roads where the resulting stiffness will start to work against you - in all fairness, the manual does warn you about this effect.
The shock absorber adjusters are labelled as Ferrari parts (with associated Ferrari pricing) because I believe they are the same as Ferrari 355, the adjustable suspension mechanism having been lifted from the 355 when the 3200GT was launched.
I hope this helps.