Problems Installing Snow Leopard from Disk

A friend just gave me his old Mac Mini (only 2-3 years old.) I turned on the mac mini and waited for it to load. I waited about 10 minutes or so and still only the white screen showed, no apple logo or anything. I quickly decided that I wanted this to be "from scratch," so I put in the Snow Leopard disk and it loaded within 2 minutes. I went through all the steps (about an hour and a half), and finally saw "Time Remaining: about one minute left." Then the numbers started going down (Time Remaining: About -4 Minutes.) I left it running for a while and came back, moved the HDMI cable behind my entertainment center, while everything was plugged in. I didn't think any harm would come from doing this, just wanted things to be a little less messy. Well, the power cord for the mac mini was accidentally knocked out! I turned the mini back on and it again took very long to load. Once the installation started up again, it didn't list a hard drive for me to select to install the OS to. After waiting 30 minutes or so for something to show up (thought it might be loading), I pressed the power button, computer reset, and the disc popped out. I obviously screwed something up majorly when the power cord was accidentally knocked out. It is currently turned off until I find someway to fix my problems. HELP!

Just start over. Boot from the install disc by holding down the C key. Or you can hold down the OPTION key, then select the drive from the list that pops up.

EDIT: I am a bit concerned that it took so long to install. Was your friend having any problems with it before he gave it to you?

Nope, not that I know of. I have a wireless keyboard and mouse, so Im not sure if the Mini will recognize me pressing the C key. After a few minutes of holding the C key down the Apple logo and a grey spinning circle below it. After 2-3 minutes I am asked to choose a language. After the next few screens I am told to select a disk to install Mac OS X to, but there are no disks to select, nothing at all. This is where I get stuck and am forced to do a hard shutdown. Any advice?

No drive to install to? Either the drive is toasted, or you have a more critical failure, like the logic board.

Really? A fried harddrive from unplugging it while it was running? Is this an expensive fix? Im kicking myself right now, if only I had waiting until the install was done to move hat HDMI cable over 2 inches!!!

No. For as long as it was taking you to install, that drive was already fried. Popping the HDMI cable would have had zero impact on it.

You should run the Apple Hardware Test to make sure the logic board and other hardware tests OK.
The Apple Hardware Test

Also read this for drive testing options:
Resolve startup issues and perform disk maintenance with Disk Utility and fsck