Samsung Captivate not booting after root I recently installed Clockwork Recovery, and used that to install CyanogenMod 10 on my Captivate using this guide. Every thing worked perfectly. 

I backed up the stock ROM. Then I realized that my contacts and data were gone in the new ROM. I read in an article that you could restore the data part of the backup using the recovery in Nandroid, so I recovered it. But when I rebooted, the phone got stuck at the spinning CyaogenMod screen.

I made the problem worse by trying to flash the mtkareys zImage kernel, and repeated the process described in the guide. Now, I can't get into recovery mode nor the Android (CyaogenMod) OS. 

I suspect that I would need to use some tool that uses my Ubuntu 12.04 or Windows 7 PC, because the only resource that I have access to is download mode in the phone. Nothing else seems to work. Please understand that I am a beginner who has never even written an app for android (I do have the sdk installed though).
What Izzy said is correct. Also, never give up! I found that there is always something you can do to unbrick your captivate. 

The xda-developers wiki page is very useful in this regard. The sections about bricking and unbricking should be very useful. While it's very difficult to enumerate all possible ways of unbricking here, understanding the nature of the brick (soft or hard) and looking for alternative ways in the forums is always helpful. The techniques that I know of so far are: 


The 3-button method and its variations, as listed in the wiki page above. (There are variations, just search for them in xda-developers)
Using your PC and the USB cable. Izzy's answer is one such method. 
Hardware solutions that involve cutting a USB cable or using resistors. When you have a hard brick, you may have to resort to these. 


Good luck and don't stop searching! There is always a way to unbrick your Captivate.