Why does my pc shut off while playing games?

I would have to guess its your PSU since the video card recommends 500W you are just at the "edge" of usable power. If you have a "no name" or a "factory" machine (with their PSU) Your defiantly on the "edge"! True you have 500W and the card calls for 500W, however, even the better PSUs are "certified 80" That means you can run "comfortably" at 80% of the listed wattage. In your case 80% of 500W is 400W. "pushing" or forcing your PSU to run at near top wattage all the time is certainly not good for it. Then to, if it has a safety Thermal couple it will kick in (or should I say "kick out') if things get to hot. This is one of the reasons for the "rule of thumb" that is: what ever your video card recommended wattage is, add 20%. In your case, 500W + 20% = 600W. That's a good rule of thumb....use it!

It seems to me that your computer requires more power. When component's are stressed, they require a bit more power so they're stable. You have a CPU which has a 125W draw on the PSU , which is quite a lot. The GIGABYTE HD 7950 is also very power hungry , it requires a lot of power to deliver a lot of performance. I would suggest you get a new PSU. I would suggest you get a 600W or higher. I would also recommend good reliable brands which are Corsair , Fractal , XFX , Bitfenix (not released yet) , Seasonic , OCZ and Rosewill. Anything that is 80% bronze efficient is good , anything which isn't , isn't an option because it could very well blow up and fry all your components!

Could be heat related could be power supply related.

It's either the low wattage on the PSU or overheating

It is more than likely overheating. Use programs like speedfan to monitor the temps.