You gave no details of your system, just what is it and OS, etc.. You gave some details but even that leads only to guesses. In such cases when it comes to PSU issues, just replace it. A 20-pin PSU connector suggest an older system. If in fact it is an older system, then more than likely it saw better days. PSU do age it may have given up the ghost, so replace. remove and/or read the decal sticker on or look by your system info what is available for your model#, if OEM, like Dell, Gateway, etc.. The cost is far cheaper than you think but again you may what to think replacement of the whole is more than a PSU problem, it maybe the mtbrd. and thus again replace. I wish I could offer more but its limited with the details you offered. For now, that may cover what is need to fix your problem.
So my computer recently started displaying no picture on its monitor. However, when I turn on the computer it sounds like not every device is powering up. So I opened up the computer and tested every connector on the power supply with a multimeter. I read that when I short out pins 15 and 16, the fan is supposed to start to spin, but it does not. However, with these two pins shorted out, I was able to take voltage measurements of the pins on every other plug, and the measurements fell within the tolerance ranges. Once I plug the 20-pin plug back into the the Mobo and power on the computer, the fan starts to spin but the CPU is so quiet that I feel like everything is not powering up. What does this mean? Where do I stand?