What kind of power supply does this machine has? Does it even has the necessary connector to power up a gaming card? Or enough power? I'm guessing no, but it's just a guess since the spec sheet doesn't even mention power supply.

That all depends a lot on what games he wants to play. Some are much more demanding on the GPU than on the CPU. As for the PSU (yeaaahh you gotta love TLAs :)), a decent modern card roughly in that price range would be a Geforce 9600GT based card, it swallows almost 200W alone when playing. Computers come sometimes with PSUs that are just able to power the stuff it came with and you would run into problems. There's a sticker on the PSU, stating how much power in Watts it has (mostly meaningless) and a chart with the Amperes it can deliver on the different lines. Alas I don't know what is healthy on current computers but I guess it should have two +12V rails with at least 20 A and more than 22A on the +3.3V and +5V lines each.

The PSU is 250 watts There is a picture of it on the web site That's gonna have a hard enough job running what you got, never mind any extras :(

I am pretty sure that 250 watts is too little 

That the GF9600GT takes 200W and your PSU has 250W doesn't mean that it works. :) The rest of the machine, namely the CPU takes most of the 250W for itself and there's not much left for a GPU on a video card. However, a slightly outdated GF7600GS (not GT) is more than sufficient for these games and takes only 40-55W max. (caution: that's an exception, for example GF5xxx and GF6xxx cards need much more power!) but even that might work or not. Another problem is that these cards are not on the shelves anymore. An overchallenged and dying PSU can fry the mobo and/or other parts when it croaks.

Ah yes.. didn't think about that part. What exactly is the onboard chip? And what games are not running? Might be that a specific game requires e.g. a higher pixel shader version than the chip supports, but more often than not the game would give you a corresponding error message. If you didn't install a very old DirectX 9 build and have the latest graphics driver for that chip installed, missing hardware functions could be the culprit.

go to this link an select what game you have an it will tell you why your game wont load