As already stated, you really do need a dedicated GPU for most games.
But that doesnt mean you need to spend a fortune on a high End or Enthusiast card either.
A card like the ATI one is only $50. Newegg.com - PowerColor AX4650 1GBK3-H Radeon HD 4650 1GB 128-bit DDR3 PCI Express 2.0 x16 HDCP Ready Video Card
I know it is a couple generations old, but do not let that sway you.
For the price, it still offers very good performance, and especially at the resolution you have listed in your specs.
At 1024x768, this little card should be able to run most games at fairly high settings even.
It would make a considerable difference in what Games you can play, and the overall quality of them.
Another good point is that it draws power from the PCI slot, and doesnt require that much. So your current Power Supply should be fine with it.
There are other options,, of course, but this is certainly one worth considering IMHO.