Hey. That speaker connected to the motherboard is really just for dos use and only comes on if you have no sound installed on your PC via a sound card (beeps when none is there when you select stuff).
But you do have a sound card built into that motherboard which will be activated (just just have no external speakers connected to hear it):
AI Audio (8-channel Audio)
So that is why the onboard motherboard speaker wont be working while running the computer (see a volume control on the task bar on the right? This shows you have souind installed)
So you'll have to buy an external set of speakers to fit into the motherboard socket (green for stereo) linked to the sound card. Because that is 8 channel, you could also get a really good speaker arrangement going if you wanted for surround games. So if you wanted to use the one on the motherboard for sound, I'm afraid thats not really possible...
You could buy a cheap stereo set if you wern't bothered about sound. Just make sure they are multimedia speakers with 3.5mm jacks
I hope this helps