I honestly do not know much about computer hardware. I apologize in advance if this post sounds illogical or impossible.
I have a rather old desktop, hp a6040n, in which I plan to acquire a graphics card to run some modern games. The recommended requirements of said games are shader model 3.0, 512 MB, GeForce 8800/ATI X1900. The current power supply is at 250 Watts. I understand that I probably need to get a power supply unit as well. Perhaps, a new system is a better idea, but I am not prepared to shell out for another system at the moment. Which graphics cards and power supply that are currently on the market would you recommend, if any?
Should be a huge upgrade for you. I would also suggest getting a better cpu in the near future and more ram. 2 more 1Gb of DDR2 Ram should be around $40 and a better CPU around $120, you have to get a cpu for your socket which is LGA775.
Total: $180 (Before Shipping/tax)
You should be able to use an low-to-mid-level AMD Radeon HD based board without upgrading your power supply. The 7750 models are the most powerful that only draw power through the PCIe slot - no need for the 8-pin power cables.
Another option (which is quickly disappearing) is the Radeon 6750 models with GDDR5 memory. About the same price as the 7750, likely a little slower, but also plug into a single slot without extra power cables required.
The 6750 boards are specified as PCIe 2.1 compatible, which I suppose would be more likely to work.
Either one will set you back a bit over $100 (US) and far exceed the specs you provided. You will probably be limited by your CPU performance rather than the video card... (is that a Core 2 or a Pentium?)