There are three main reasons you should lower your tyre pressures when driving your 4X4 off-road.
The first and most important reason is that it improves traction. Lowering your tyre pressure increases your tyres footprint which increases the amount of rubber touching the ground. More rubber = More traction.
Secondly, it improves the rideability and comfort of your 4x4 when driving on rough terrain. Lower pressure in your tyres means that there's more cushioning between the road and your vehicle.
Thirdly when driving off-road you are likely to come across lots of sharp objects on the tracks (sticks and stones, etc.). Lowering your tyre pressure means that your tyres are less likely to get a puncture if you happen to run over something sharp as there is more flexibility in the rubber.
Different pressures for different terrains.
There are many different theories on what pressure you should be running on what surface.
It all comes down to what vehicle you are driving, what surface you are on and what works best for you.
A general rule of thumb is as follows, depending on the size of tyres and weight of the vehicle, etc.
Keep in mind this is an approximate guide only and can be different for your vehicle.
Gravel roads. An approximate drop of about 1/4 of your normal highway tyre pressure is a good starting point for gravel roads
Mud and rough terrain. An approximate drop of about 1/3-1/2PSI of your normal highway tyre pressure would be a fair indicator of where you want to be at driving these terrains.
Sand terrain. An approximate drop of about 1/2 of your normal highway tyre pressure is where you want to be at driving on sand, keeping in mind if you have bead locks you can possibly take it down a bit more.
Again the correct tyre pressure does vary slightly from one vehicle to another and there are many different factors to take into consideration to find the correct pressure for your vehicle and the terrain you are driving on.
Always remember to re-inflate your tyres as soon as you get back on-road!!!!