Seemingly, my computer (main desktop) does not send information about the sound of what goes on the left speaker to the speakers. I tried the speakers on my laptop, and there the left speaker works fine.

I am wondering if it is a hardware or a software issue. The only reason for my concern, is that it means I will have to bug my mom to buy the CPU she owns me - because I already have the motherboard, etc.

But if it is software, it should be easy to fix, I guess. The soundcard is an onBoard soundcard, so I am not sure what it is - but the left speaker used to work.

Any help - or suggestions to allocate the problem?

Did you use the same audio (stereo) cable when you tested on the laptop?
Did you try another pair of speakers on the desktop?

If so, sounds like the sound card has lost it.

Another interesting fact is that when I use the secondary plug I have as sound output (there is an extra plug in my case, so it is hugged up to my motherboard), it works fine. And yes, I did try with the same cable.

...Then it's not a software issue, and the "onboard soundcard" is fine.
So it must be "hardware"...
So you need to check your connections and cables.
See this as a puzzle: Take a methodical and logical approach, and you'll soon have the fault located.