Q: mac mini won't start up again...

- Try anther monitor and cable unless you know the monitor and cable works with another computer

- Try resetting the SMC and NVRAM/PRAM

Intel-based Macs: Resetting the System Management Controller (SMC)

About NVRAM and PRAM       

- Try starting in Safe Mode

OS X: What is Safe Boot, Safe Mode?

- Boot to Recovery and repair the startup disk

OS X: About OS X Recovery - Apple Support

- If it is repairable reinstall the OSX

How to reinstall OS X on your Mac - Apple Support

- If you do not have a backup use disk utility to restore the internal disk to an external disk so you can try to recover data.

Then format the startup disk and do a fresh install of OSX

Note Recovery requires Lion or later. If earlier you will have to boot from Install DVD vice boot to Recovery

What model Mini and OSX version?

I had the same thing happen to me yesterday... So I took to the apple store and then it wouldn't start up in safe mode..  So they tried resetting the P-ram and then it started right up in the apple store...  Took the computer back to the office and then wouldn't start up again until I reset the p-ram again...   Used the computer yesterday afternoon just fine then put it to sleep last night and the morning had to reset the p-ram again to get get it to start up...  Funny things are happening... 

-Chad