My computer is a Slim version and I need a new ATI graphics card

are you looking to play games or just watch DVDs/stream video/office work?

If you need more power than the HD4350 the something like the HD5550 or 5570 would be the better of the 'new cards' that would fit the bill;

http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Is this a slim card? Because this is the size of my computer tower:
The back side:
Front Side:

Yes, that is a low-profile ready card (will fit your slimline). What you need to do with that card is remove the pre-installed (full-size) metal bracket and replace it with the two included low-profile brackets (see them in the fourth image). Very easy to do, btw.

-Wolf sends