Maintaining traction is the single most important part of mountain biking. Having fun on the dirt is directly related to staying upright, regardless of how fast you want to go. And you won't stay upright, unless you keep the rubber gripped to the dirt. When I first started mountain biking, I pumped my tires up to 40psi and went out into the woods bouncing and crashing all over the place. I thought that since I had this fancy front shock and big tires that I should be able to roll over roots no problem. Needless to say all the crashing made me not like the dirt very much. Then a friend went with me and taught me a few things about tire pressure.
When you are on the trails, you must be able to grip your riding surface. That looks like a few things depending on your terrain. If its muddy, you need traction to maintain forward motion and to not slide in the turns. In the dry dirt, you need some level of grip so your front tire doesn't wash out on loose dusty or sand. Over roots, you need the rear wheel to grab onto what may be a slick bark and propel you over it. If you're on rocks, you want to maintain grip on the face of a rock so that you don't slide when your weight is on that rock. Lower tire pressure accomplishes all these things.
The "Contact Patch" refers to the amount of tire you have in contact with the ground at a given moment. The more tire you have in contact with the ground, the more likely it is that you will maintain traction. A tires pressure affects how large the contact patch is; the higher the pressure, the smaller the contact patch. There is a balance though; taking your tires pressure too low will cause you to bottom out, ie: compress the tire to the rim, if you hit an obstacle hard. I weigh about 163 lbs and I like to ride my tires around 20-25psi. With much trial and error, I've found that pressure supports my weight without bottoming out, but also gives me a big contact patch on the ground for maximum traction in turns, going up hill, over roots and rocks, and through deep mud.
The more fine tuned your pressure is, the faster you will be able to go, while staying upright more easily.