As a general rule, it is best to run the maximum allowed pressure as specified on the sidewall of the tire. The tire will last longer, handle better, and will provide the best fuel economy. The only down side is that, at higher pressures, you will experience maximum road noise and/or harshness of ride.
GoWesty recommends starting at the maximum pressure as specified on the sidewall of the tire, measured at ambient temperature (referred to as "cold" on the sidewall). Then you can start reducing pressure from there if you cannot handle the harshness and/or noise—but remember that you'll be giving up tire life, handling, and fuel economy. Generally, you want to avoid dropping below about 75% of max for driving on pavement.
When off the pavement for more than just a few miles, it is a good idea to "air down." A softer tire can absorb more energy than a hard tire. By airing way down, your vehicle's floatation is improved—which can be useful in certain circumstances.
When driving on hard surfaces—like graded or gravel roads, over rocks, and stream crossings—at moderate to high speed, you will want to drop the pressure a bit. This will allow the tire's sidewall to flex a bit more than it would at full pressure. This extra flex will absorb more energy so that your suspension, body, and spine have to absorb less. But, you don't want to drop the pressure so low as to allow the sidewall to collapse entirely and bend the wheel... or worse yet, pop off the bead of the wheel and come completely off! For hard surfaces, dropping the pressure 60-70% of max psi is generally a good range.
On super soft surfaces like sand or mud, going even lower is sometimes necessary not for energy absorption, but rather to avoid getting stuck. In this situation, dropping down to 50% is about right. If you are stuck, it may be necessary to drop down to as low as 25% of max psi to get un-stuck. The reason this works is because a greatly deflated tire gets much wider at the bottom, thus increasing your footprint and distributing the weight over a larger area. That is where the increased floatation comes from, and it really, really works! It is not wise to drive around at under 50% tire pressure, though, because it might cause the tire to come off the wheel when negotiating a turn. So if you drop down to get un-stuck, you need to inflate back to at least 50% when you get un-stuck and continue on your journey.
How to calculate the pressure:
To figure out the actual psi to drop down to, just multiply the max pressure on the side of tire by the decimal equivalent of the percentage. For example, a tire that calls for a max pressure of 50psi dropping to 70% is calculated as follows: .7 x 50 = 35psi. Dropping to 30% is .3 x 50 = 15psi.
When we travel in Baja, we take along tire deflators, which make it very convenient to drop the pressure to the desired level, even in the dark. I like to carry one set adjusted to 35psi, and a second set in a different color adjusted to 15 psi.