You do realise you've opened a can of worms and now all the old barricades are being manned again?
To deal with your original question: Traction Control in VAG cars is a system to prevent the driven wheels from losing grip when accelerating. It is implemented by the ABS computer which is attached to the ABS pump and uses the ABS wheel speed sensor information. There is no extra hardware involved with TCS - it is all in software, written by Robert Bosch GmbH, makers of the ABS system.
Some have mentioned ESP - this is much more than Traction Control, and is in all ways more complicated. I'll keep to Traction Control in this post.
Well, almost. Just what is ABS? The ABS computer takes wheel speed inputs and monitors them when you put the brakes on. If one wheel is slowing down a lot sooner than the others (i.e. it's locking up), the ABS computer releases the brake pressure on that wheel by pulsing it using a solenoid pump (you can hear and feel this) to regain grip. The purpose of ABS is to keep the wheels from locking up so that you can still steer under full braking and have a chance to avoid an obstacle. It doesn't provide the absolute best stopping distance, but it's better than most drivers, I think, as it takes the road surface under the tyres into account.
So the ABS system has wheel speed sensors, brake pressure sensors and can affect the braking of each wheel, by means of the software in its dedicated computer, the ABS control unit.
The same control unit can be used to control more of the car's behaviour. You can look at TCS (the Traction Control System) as ABS-in-reverse, controlling undesirable wheelspin during acceleration. The unit can tell if the front wheels are spinning faster than the rears. The TCS software sends a signal to the ECU which cuts the power to the front wheels, keeping your tyres in contact with the road.
This is the aspect of TCS that some find troublesome. It is felt to be intrusive, and indeed on modified cars it sometimes is reported as being alarming, cutting power in dangerous situations.
My feeling is that:
- a remapped car may need the TCS software to be adjusted. Especially on a petrol car where the throttle plate must move in coordination with the fuel quantity being reduced to cut power smoothly. Remapped cars are more likely to react badly when asked to cut power in acceleration. Diesels have it easier as only fuel quantity needs to be managed.
- Good, matched tyres, suspension modification, road conditions and driver ability also play a big part in provoking the TCS system. The TCS software was written for a car in good condition with standard ECU software, standard suspension and the same tyres on all four wheels.
Poor tyres will provoke TCS, most especially when it's wet or in snow+ice.
Mismatched tyres (with different rolling radius) will provoke TCS because of the difference in wheel speed. This is why TCS should be disabled if you have to put a space-saver spare on.
Lowering suspension leads to stiffening (less travel, same weight to manage) which some drivers interpret as an improvement in handling. In most cases handling (ability to go round corners/change direction and keep a grip on the road) is made worse, and the early triggering of TCS is a symptom. The stiffer car simply cocks a wheel, losing grip on the inside of the turn, or if the car is upset by potholes or bumps or oil on the road or slippery white-line markings etc. etc.
Leadfooted driving can also provoke TCS, particularly in combination with the above.
When you turn it off, you are most likely to feel a difference on poor road surfaces (rough or slippery) where the tyres will loose grip and pull the car to one side or the other. You may also see an effect when accelerating through long corners or out of short ones, where the loss of grip will lead to understeer - sliding towards the outside of the corner.
I've gone on a bit, haven't I

I can pontificate for England given a little push. I've not covered EDL or ESP, but feel I should stop now.