If I may make a couple of suggestions based upon your budget.
If you are going all-out gaming:
Case: Zalman Z9 Mid-tower
http://www.newegg.com/Product/Product.aspx?Item=N82E16811235026
Power Supply: Rosewill Capstone 550-M
http://www.newegg.com/Product/Product.aspx?Item=N82E16817182262
Motherboard: AsRock Extreme4
http://www.newegg.com/Product/Product.aspx?Item=N82E16813157293
CPU: Intel i5-3450
http://www.newegg.com/Product/Product.aspx?Item=N82E16819116506
GPU: XFX Core AMD Radeon HD 7870ghz Edition
http://www.newegg.com/Product/Product.aspx?Item=N82E16814150604
RAM: Patriot Viper 3 1600mhz
http://www.newegg.com/Product/Product.aspx?Item=N82E16820220685
HDD: Western Digital Caviar Blue 500GB
http://www.newegg.com/Product/Product.aspx?Item=N82E16822136769
ODD: LG DVD burner
http://www.newegg.com/Product/Product.aspx?Item=N82E16827136247
OS: Windows 7 Home Premium
http://www.newegg.com/Product/Product.aspx?Item=N82E16832116986
Heatsink: SilenX 80mm
http://www.newegg.com/Product/Product.aspx?Item=N82E16835226049
TIM: GELID GC-Extreme
http://www.newegg.com/Product/Product.aspx?Item=N82E16835426020
Total before shipping: $960
Cost reductions for getting a monitor:
Lower end motherboard-
http://www.newegg.com/Product/Product.aspx?Item=N82E16813157302&name=Intel-Motherboards
Lower end GPU-
http://www.newegg.com/Product/Product.aspx?Item=N82E16814161426&name=Desktop-Graphics-Cards
and/or: Go with an AMD AM3+ socket motherboard and a AMD quad core, such as a Phenom II X4 965.
If you mostly are just playing older games like Nexon titles / Mabinogi and are looking at more casual use:
Case: Zalman Z9
http://www.newegg.com/Product/Product.aspx?Item=N82E16811235026
Power Supply: Rosewill Capstone 550-M
http://www.newegg.com/Product/Product.aspx?Item=N82E16817182262
Motherboard: ASUS F2A85-M/CSM
http://www.newegg.com/Product/Product.aspx?Item=N82E16813131881
APU: AMD A10-5800k
http://www.newegg.com/Product/Product.aspx?Item=N82E16819113280
RAM: Patriot Viper 3 1866mhz
http://www.newegg.com/Product/Product.aspx?Item=N82E16820220692
SSD: Samsung 830 128GB
http://www.newegg.com/Product/Product.aspx?Item=N82E16820147134
HDD: Western Digital Caviar Blue 500GB
http://www.newegg.com/Product/Product.aspx?Item=N82E16822136769
ODD: LG DVD burner
http://www.newegg.com/Product/Product.aspx?Item=N82E16827136247
OS: Windows 7 Home Premium
http://www.newegg.com/Product/Product.aspx?Item=N82E16832116986
Heatsink: SilenX 80mm
http://www.newegg.com/Product/Product.aspx?Item=N82E16835226049
TIM: GELID GC-Extreme
http://www.newegg.com/Product/Product.aspx?Item=N82E16835426020
Total before shipping: $762
Given you had not shown any interest in overclocking, there is no point in going with a Sandy Bridge CPU, hence my choice of an Ivy Bridge CPU. While the 560 is a good card, it is not in the league of the 660 or Radeon HD 7870, of which the 7870 is slightly faster and offers better performance / watt. Either card will max out older games, and offer stable frames per second above 30fps in newer titles, and in some over 45 frames per second.
I suggested AMD trinity for the casual build as an SSD (solid state drive) will make your system more responsive overall versus an Intel CPU, and the integrated graphics are more than powerful to play any game out there, though newer titles will require lowered settings. It is about on par with a Nvidia Geforce GTX 640/ Radeon HD 6670 (DDR3) if you want to look at discrete options which are similar. Plus, should you need more performance, you can always add a dedicated GPU down the line without issue, or upgrade to a future AMD APU. While a 7200rpm HDD is notibly faster than a 5400rpm HDD, they are nowhere near the almost-instant speeds of an SSD.
Again, if you're not overclocking, you don't need a top-end motherboard. However a good power supply is always important for the reliability of a system. And quiet aftermarket heatsinks are always nice, as is thermal paste that will last as long as you use the system.
RAM choice is made by that 8 GB is more than enough for anything today, and you can always add more later. AMD trinity loves higher-speed RAM.
Case choice for both systems is the Zalman Z9, an excellent value case offering sufficient airflow for all but the highest-end components, decent cable management options, and a bottom mounted PSU. The power supply choice for both is a Rosewill Capstone 550-M, which is based on the very reliable Superflower Golden Green, and the modular cables allow for easier cable management.
For more information on building a system should you be looking at modern gaming specifically, I would recommend checking out my guides. (under the username of Vulpesveritas) I would start with my overview on common misconceptions;
http://mwomercs.com/forums/topic/8360-guide-hardware-mythbusters-an-in-depth-hardware-guide/
One other thing to consider, for the price of a Nvidia Geforce GTX 560, you can get an AMD Radeon HD 7850, which is on average around 30% faster, and uses about 50 watts less power, even should you primarily go with another build.
http://www.newegg.com/Product/Product.aspx?Item=N82E16814161426&name=Desktop-Graphics-Cards