Recommended card: Asus R9-270
http://pcpartpicker.com/part/asus-video-card-r9270dc2oc2gd5
The Asus R9-270 is about the same price right now (after rebate it's $151) as the R9-265 and the Asus model overclocks well (about 1100MHz) thus it makes no sense to get the R9-265:
http://www.techpowerup.com/reviews/ASUS/R9_270_Direct_Cu_II_OC/29.html
*FREE GAMES:
The R9-270 also has two free game codes (the SILVER list). Narrow it down then look at current Steam prices if you want. For example, "Sniper Elite III" and "Tomb Raider".
Other stuff:
1) You can read your motherboard manual if you want under any VIDEO or GRAPHICS sections to see what it says but I doubt there's a need.
2) I recommend updating your BIOS if you haven't done so (not for the graphics card, just to ensure stability):
http://www.gigabyte.com/products/product-page.aspx?pid=4716#bios
3) run MEMTEST at some point to ensure it's stable. www.memtest.org (Boot Order in the BIOS needs the DVD or USB drive as applicable to be first, then your Windows drive second otherwise it just boots straight to Windows. Run a FULL PASS which is roughly 30 minutes for 8GB. This test the DDR3 memory sticks.)
4) Also, update your audio, chipset, and network drivers:
http://www.gigabyte.com/products/product-page.aspx?pid=4716#driver