Keep in mind that the installer has to boot your Mac mini. That means the DVD has to be correctly created as a bootable DVD. It ALSO means that your DVD drive has to be in (reasonably) good condition. Try running a cleaning disk on your drive (or even blowing in the slot with a "can-o-air" can help)
(I used to burn a lot of Leopard installer DVDs, and a good percentage (maybe one in three) would be bad copies. It's somehow connected with how the DVD boots, and is quite common for the "bootable" part (which is the main thing you need to work on an installer disk) to not work (no boot, or shows an error of some kind. I learned pretty quickly the clues to know when a DVD wasn't going to work. Again, OS X Leopard, for some reason, is the most tricky to get it all working at the same time. I dumped quite a few dual-layer DVDs in the trash
What happens when you try booting directly from the installer DVD?
Restart, holding the C key. That's the keyboard shortcut to force your Mac to attempt booting from the DVD drive.
If that doesn't work, restart, holding the Option key. You should see any bootable partitions as choices. Click on the installer (probably named "MacOS X Installer", or similar), then press enter. If you DON'T see any choice for the installer on that Option-boot screen (wait for a minute for a slow response to read the DVD), then it's not bootable (a poor job creating that DVD)
If it still refuses to boot to the installer, then it's likely a faulty DVD. If it is a copy (not the original, Apple commercial Leopard installer on a black and purple DVD) then you might have some luck by copying (restoring) that DVD to a separate partition on an external hard drive. The Mac's Disk Utility might work, when you "Restore" the DVD to a partition on an external. 8GB is a good size for that partition. That will give you a (noticeably) faster install than the DVD, and will often be less fragile than the DVD disk. I hope that works for you.
