Put a new hard drive into my Mac Mini. How do I get the software back on it?
If you have your old disk, you can simply connect it by USB or Firewire or whatever external box you have and drag your applications from your old drive to the new one.
Most Mac applications do not need installation and can be simply copied/moved. Notable exceptions are Office and iWork, which you will have to reinstall from the original media (DVD or download, depending on the case). You might have to reinstall some other apps, depending on what you use.
If you have a Time Machine backup it can restore both your apps and your settings. Best possible solution IMHO.
If the original disc is still intact, put it into a USB-enclosure, start up the Mac with the ALT (=Option) key down, and choose to boot up from the old disc. Download SuperDuper! from Shirt Pocket software and use that to clone your drive (or you could use Carbon Copy Cloner if you prefer) and that's it! Next time you do something like this, start with the original drive still in the Mac and use the new disc as the external drive, which will save you some time and grief.
If you have been making backups in a manner that allows for performing a bare-metal restore, you can simply restore the latest backup. Otherwise, you will need to install your OS, then restore your latest filesystem backup. If you don't have a filesystem backup either, you will need to reinstall everything from your original media.
which software? have you made backup