My computer just shuts off for no apparent reason. If I boot it in the morning after its been sitting all night, it will stay on for about 30 minutes, but after that the time I have on it gets shorter and shorter. I took the casing off and looked inside. There is a fan mounted I believe to the motherboard, its behind the powerhouse, or where the power cord plugs in. If this fan is not running does the motherboard overheat and shut down?

Processor is overheating. the fan you're referring to is the case fan

ASPIRE ATXB4KLW-AL Antec 400W
ASUS A8N-SLI Deluxe
AMD 3700+ San Diego
Corsair 1GB 2x512
eVGA 6600GT
250GB Hard Drive
Sound Blaster Audigy
Windows XP HOME W/SP2

Yes all your fans should be turning, and this could lead to a shut down, those fans can be replaced however if you are not familiar with working around electricity and no one you know does its time for a new power supply don’t go cheap when your shopping maybe take a look at an ANTEC power supply, when checking for the wattage don’t push the load of it more than 80%
all that means is its better to get one a little bigger than you need.....

Keep the old stuff running

Assuming any warranty you had with Dell has expired, I also recommend replacing the power supply. I haven't seen inside many Dells recently, but if your current power supply has a specs sticker on the side, use that to pick out the new one. The most important specs are Amps at +12V and overall power. The +12V current rating is especially important with newer systems because lots of power supplies out there are inadequate in that regard despite having a ridiculously high overall power rating.