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.
bicycles.stackexchange.com/users/167/Adam Franco
Generally run what the tire says.
If you know the terrain is slick rock or really loose and will require more contact between the tire and the surface then make them a bit softer.
This will increase the chance of pinch flats though so be careful.
I almost always run my front tire a bit softer than my back since most of my weight is on the back tire.
Just something I've found that I prefer.
bicycles.stackexchange.com/users/30/curtismchale
It really depends on that the trails are like as well as your weight.
If you are riding really flowy trails with few obstacles, jumps, or drops, you can probably ride a bit under 25psi if you are light.
THis will give you significantly more traction and control.
If you are riding trails with jumps and drops, you will want to air it up more.
With rockier trails, you will want to ride at a higher psi to avoid pinch flats.
These are the kinds of trails that I typically ride and I usually go with 30 - 35 psi in the front and 35 - 40 psi in the back.
bicycles.stackexchange.com/users/67/Geoff
I would say that you should run the lowest pressure you can without the tyre rolling off the rim on corners or burping on rocks.
The assumption that harder tyres have less rolling resistance is not necessarily true, here is a study showing the opposite.
[updated link, if it breaks again, search Google for "Rolling Resistance Eng Illustrated".
The document was originally written by Schwalbe Tires]
bicycles.stackexchange.com/users/126/Richard Tasker
I agree with the other answers, but my strategy is to run the lowest pressure I can while consistently avoiding pinch flats.
There's too many variables to come up with a general answer (bodyweight, trail type, tire manufacturer), but I find that something in the mid-30's is safe for me (200+lbs, xc race courses).
If you want to play it safe, find a safe pressure and add 2-3 psi.
Also, always check your tire pressure, taking the few minutes before most rides keeps you riding longer.
bicycles.stackexchange.com/users/153/Dana the Sane
A tip my friend showed me at the weekend was to take to take a wheel and find a pointy rock.
Lean all your weight gently onto the wheel.
If the wheel doesn't hit the pointy rock then you're not likely to suffer from a pinch flat.
If the rock easily touches the rim then the pressure is too low.
This should account for a number of critical variables such as the volume of the tyre and the weight of the rider and the reasoning that you'll hit the rock at speed or on one wheel only (normally each wheel only takes 1/2 your weight).
bicycles.stackexchange.com/users/147/davefiddes
Some very good answers, but I think it's also worth saying that the sidewall gives a maximum not suggested pressure on all the bike tyres I've seen.
The only time I ever run my mountain bike tyres at their maximum is if I'm riding on the road and in a hurry.
bicycles.stackexchange.com/users/825/Duncan McGregor
None of the answers mention tubeless tires.
A tubeless tires has no tube so no tube to pinch.
The primary purpose of tubeless is for lower pressure.
Need both tubeless rims (wheels) and tubeless tires.
Can get conversion kits for regular rims.
Tubeless comes in fully tubeless and tubeless ready.
With tubeless ready you need to use a liquid sealant.
Tubeless ready is a more common.
bicycles.stackexchange.com/users/7785/paparazzo
Should I run my mountain bike at the suggested PSI/Bar on the tire/rim?  Or should I run it a bit lower?  Or is it more about conditions?  Should I run lower if I am running on softer terrain?
What pressure should I run my Mountain Bike tires at?
bicycles.stackexchange.com
<mountain-bike><tire>
bicycles.stackexchange.com/questions/471
bicycles.stackexchange.com/users/97/Dan McClain
There is no general rule for tire pressure on a mountain bike. It really depends on the type of trail you are riding, rider's body weight and tire manufacturer. A general tip is lower pressure where more grip is required.  However, care must be taken not to go too low with the pressure. Tire pressure may depend on a number of factors, including grip, slope and road quality. In general, more steep and even routes require lower pressure, while the opposite applies for rocky trails, or roads with obstacles. While there is not a single answer, identifying a safe pressure for your bike and adding 2-3 psi might be a good idea. You could also try riding with the lowest possible pressure that does not affect the tire rim or corners, keeping in mind the increased possibility of pinch flats. Finally, you could follow the manufacturer's instructions on the tire.
Tire pressure may depend on a number of factors, including grip, slope and road quality. In general, more steep and even routes require lower pressure, while the opposite applies for rocky trails, or roads with obstacles. While there is not a single comprehensive answer, identifying a safe pressure for your bike and adding 2-3 psi might be a good idea. Consider following the manufacturer's instructions on the tire. You could try riding with the lowest possible pressure that does not affect the tire rim or corners, keeping in mind the increased possibility of pinch flats.