|Yup, the blue board is indeed the motherboard, the one that says Gigabyte on it.|
So this is your motherboard?
What operating system are you using?
You can often set each of the power buttons to do something different than the norm, such as putting it into 'Sleep Mode' instead of actually shutting it down, although this wouldn't change what clicking Shut Down in the start menu does.
Like Derek said, your best bet at this point is probably just to make sure all of the connectors are in place.
Quick question, do any lights appear on when you try to shut it down? I.e. is the power light on, flashing, off? Also, do you hear any beeps or other noises (besides the fans running)?
Have you tried using a different monitor? Also, do you get a signal if you disconnect the monitor completely then reconnect it?
Does the system shut off when you hold the power button down for about 5 seconds?
CPU: AMD FX-8350 - 8 Core @ 4.0 GHz
RAM: 2x4 GB @ 1600 MHz
Mobo: MSI 970A-G43
PSU: Corsair CSX 600W
Graphics Card: Radeon 7870 GHz Edition - 2 GB GDDR5
OS: Windows 8.1