If the system is running fine now, then nothing has been damaged. The danger is typically when they get hot, they can pop taking other components out with them. If you NEED to use the computer in the meantime, I'd refrain from straining it too hard (gaming for example). Assuming that popping sound is coming from the PSU, I personally wouldn't use the system until I got a new PSU in there.
If you are in the US and on a tight budget, here is a great deal on an EVGA 500B unit $29.99 with $20 mail in rebate
This one should be fine for your system assuming you are not overclocking. http://pcpartpicker.com/part/evga-power-supply-100b10500kr