Will this RAM work with this mobo?

You didn't link to a specific mobo. However, the RAM you linked is good RAM and works on most modern configurations (DDR3, 1333mhz, 1.5V, CAS latency 9). You might want to get the same RAM in a 1600mhz setup because it will be about the same price and a little faster.

You will almost certainly *not* notice the difference between 16GB and 8GB of RAM today unless you deal with specific applications that use a large amount of RAM. You can buy a 2x4GB kit now and add another 2x4GB kit later if you feel like you need it (and if you buy a mobo that has four slots, which most do).

These the correct links... Peiggs's links don't direct you correctly unless you copy them to a text file, then copy them in text to the browser:

RAM G.SKILL Ripjaws Series 16GB (4 x 4GB) 240-Pin DDR3 SDRAM DDR3 1333 (PC3 10666)
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
Mobo ASRock 970 EXTREME4 AM3+ AMD 970
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Click on the 'Details' tab to see the RAM information: Memory Standard DDR3 2100(OC) / 1866(OC) / 1800(OC) / 1600(OC) / 1333 / 1066 / 800

ASRock website, product; ASRock 970 Extreme4 AM3+ motherboard Specifications;
http://asrock.com/mb/overview.asp?Model=970%20Extreme4&...

Yes that 1333Mhz RAM works with the ASRock 970 EXTREME4 AM3+ Motherboard. Also RAM from 1600(OC) to 2100(OC) should work on your motherboard but at 1333MHz... or overclocked at full potential.. drawback is it will probably work hot overclocked so either install fan coolers or stick to 1333MHz. Or get Cas Latency 7 G.Skill RAM for $65.00
G.SKILL Ripjaws Series 8GB (2 x 4GB) 240-Pin DDR3 SDRAM DDR3 1333 (PC3 10666)

Or, I would suggest you spend $30.00 more on a better motherboard, the ASRock 990FX Extreme3 with support for 1866MHz RAM for $140.00
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

RAM to go with the ASRock 990FX Extreme3:
CORSAIR Vengeance 8GB (2 x 4GB) 240-Pin DDR3 SDRAM DDR3 1866 (PC3 15000) $80.00
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
Or G.SKILL Sniper Gaming Series 12GB (3 x 4GB) 240-Pin DDR3 SDRAM DDR3 1600 (PC3 12800) 12GB for $80.00
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Thank you very much guys!

first of all, don't get a 3x4GB kit. That's for triple channel motherboards. Running three DIM of RAM will *significantly* reduce your performance.

Also, 1600mhz is the sweet spot. It's the same price as 1333mhz and upgrading to 1866 is expensive and gives very little performance improvements. I wouldn't pay for CAS latency 7, either.