What pressure should I run my Mountain Bike tires at?
Tire pressure is generally a trade-off between three things:


Rolling resistance (more pressure == rolls easier)
Pinch flat resistance (more pressure == less chance for the tube to tear when a rock squashes the tire toward the rim)
Grip (with less pressure, the tire can conform better to rocks, roots, and other terrain giving a larger contact surface)


If you are riding terrain where grip isn't too much of an issue (flat or low-angle dirt and dry rocks for example), then higher pressure will keep you going fast without pinch flatting.

If you are riding steep downhills and/or on slippery open rock outcroppings, grip will be more important than rolling resistance, so use as little pressure as won't pinch-flat quickly.

There is a 4th trade-off as well: weight. You can buy double-sidewall downhill tires that allow you to run very low pressures (such as 25psi) without any danger of pinch-flatting, but these can weigh almost double what normal cross-country tires do.

As mentioned in other answers, your tires don't have to be the same pressure. Usually grip is more important on the front tire since most of your weight is on it while descending difficult downhill sections. Similarly, when riding on flat or climbing, most of your weight is over the back tire, so rolling resistance and pinch-flat resistance are more important for the back. 

I personally ride with a low-pressure (30psi in a tire rated 35-65psi) double-sidewall downhill tire on the front and with a higher pressure (50psi in a tire rated 45-65psi) cross-country tire on the rear. I sacrifice some weight, but otherwise get the best of both worlds: fabulous grip going down and easy rolling on the flats and climbs.

Use your best judgment in going below the rated pressure of the tire, as pinch flats become more likely. If you go below the rated pressure, be sure to test it out on hard impacts (such as a rock-corner) on easy terrain before throwing yourself at high speed down a mountain where a blow-out would be catastrophic.