The motherboard and power supply is more what you have to be worried about for running a graphics card.
But to answer your question, yes you can put an Nvidia or AMD (Gigabyte is a brand) graphics card in. Personally I'm an Nvidia person but if your looking for a lower price range card that'll perform decently I've heard the 7750 or 7770 from AMD are very good for their price.
You will need to get a better power supply and I recommend that no one ever uses the power supply that comes with a case, Seasonic are my pick for power supplies. http://www.pp.co.nz/products.php?pp_id=AA38312