Reformatting a Mac Mini

You need some kind of installation medium to reformat the Mac, as reformatting it from inside a running Mac OS X will not work, as the running OS needs a place to run from, thus it can't be formatted (and been taken away its ground to run from).
If you have access to an external optical disk drive (ODD) or a Windows computer with an internal ODD, get the 29 USD Mac OS X 10.6 Snow Leopard Upgrade DVD and use Remote Disk Sharing.
Also know, that that Mac originally had an ODD.

MacBook Air, Mac mini: How to use Remote Disc to share DVDs or CDs from a Mac or Windows-based computer

 

I suggest purchasing lion on another mac machine and then use the downloaded medium to create yourself a USB bootable drive from which you can install lion freshly.

If you don't want lion, then you'll have to purchase snow leopard on disc to either install from an external ODD or use a USB boot of SL.

Alternatively, you can purchase recovery USB sticks for either OS on eBay for relatively cheap, although I don't know the legality of what the sellers are doing. It could be considered copyright infringement because they are essentially selling the operating system, on a usb drive.

 

Couldn't the someone you got the Mac Mini from give you the username and password so you can reformat this?

Seems odd that they wouldn't have either wiped it already or have provided you with the username and password for you to be able to.

 

suggest grabbing a copy on another Mac from the App Store and do a fresh install of Lion with the USB or DVD Good Luck