I have a mac mini (model that has no DVD slot with 4gb ram & 500gb HD) and a mac-mini-mac-os-x-install DVD disc. I tried to install os on it with an external DVD drive but it booted up with an apple icon in the center of the screen and nothing else. Please help. Thanks in advance.
A Mac Mini OSX Install DVD supports the particular release of Mac Mini with which it was released, not all Mac Minis. The fact that your Mac Mini does not have a DVD slot suggests that it likely did not ship with that install DVD. Additionally, in general, Apple hardware does not support installing versions of Mac OS X older than the version current when the hardware was released.
This may be the source of your problem.