How do I rebuild without my original software disk?

I need to rebuild my mac mini for many reasons. However, I do not still have the original disk with OSX on it. I do have an external hard-drive (dont know if that will be helpful or not). Is there a way that I can download OSX or can I transfer it to my external hard drive and put it back on? Furthermore, I would love to upgrade to something like snow leopard but can't afford it. Any suggestions would be greatly appreciated!

No. You have to spend a buck and buy whatever disc you are after. Snow Leopard is US$29.00.

We do not discuss piracy here.

You can't legally download OS X (other than incremental updates). To install those you first need a purchased copy of the initial release: To install the recently released 10.6.6 release for example you need to have purchased a previous version of 10.6.

You could transfer your current installation to the external drive, then erase the internal drive, then transfer everything back from the external drive to the internal drive. That's only a good idea if your current installation is working well and not having problems. If you "clone"an installation with problems the problems are still there. If you can explain why you need to rebuild perhaps we can give better advice.

BTW what are your machine and OS specs? With that we can provide much better answers. Depending upon what OS you are running now upgrading to Snow Leopard may be as little as about $30.00.