By looking at that picture that was given, I would have to guess that you need to get a "low profile" graphics card. I basically did that back when I upgraded my Dell a few years back. A normal card may be too tall to fit. No worries though, that's why they make the "low profile" cards.
My suggestion would be to check this page:
Newegg.com - low profile graphics card
Keep in mind that you have to check the cards specs for required wattage and output type. Like a card that requires a minimum power supply of 400W may not work for you. The output port would be like DVI-I or HDMI, some of the "low profile" cards have both. You just don't want to buy a card that has an HDMI output and you don't have an HDMI input on your monitor.
So look at that list and narrow down the cards by your price range first.
Then check the specs of the ones you have narrowed down. Checking for the minimum power supply wattage. Once you have those narrowed down, probably to a handful, go and check what type of video output port it offers.
DVI outputs High Def video but no audio
HDMI outputs both High Def video and audio
Other options may be VGA which is only analog
Hope this helps.
PS - when I upgraded my slim DELL it had a 550W power supply in it. I choose to go with the "low profile" NVIDIA GeFORCE 8400 GS with 512MB or RAM. It was an improvement over the onboard Intel graphics but I still had to play most games at low settings. Some at medium.