There are two systems controlled by the little button on the console. One is traction control. It is designed to limit wheelspin during acceleration. It works by modifying engine timing, throttle application, and brake application. The second function is a stability control system. There are yaw sensors in the car that are compared to steering wheel position, etc. The computer determines if there is more car rotation than anticipated. If there is, the system selectively applies brakes to the appropriate wheels to bring the car back to a stable position. I think it also modulates throttle. These descriptions may not be technically correct, but you get the idea.
I leave all the systems on for normal driving. Why disable safety systems for normal driving? I turn both off for autocross. I want to be able to control car rotation without having to fight the computer system. I use the competition mode at the drag strip. It turns off the traction control but leaves the stability control on. I want some wheelspin (only a little!) at the strip, but I want the stability control. It might keep me off the wall if I screw up.
Does this help?