Traction control/ Auto LSD ?

I have a 09 4x4 DC with the button that has the squiggly lines under a vehicle. I think its traction control. If you hit it the indicator light with the squiggly lines OFF comes on as well as the AUTO LSD indicator light.
If you hold the button for 3 sec, just the squiggly line indicator light comes on.
Can some one explain what these mean and what conditions to use them? The manual if very confusing and the service guy at the dealer was even more confusing.

I've noticed that while driving, if the wheels slip there is an audible beeping alarm and the squiggly line indicator light comes on.

Thanks, John