I was upgrading from 12.04 to 12.10 when the battery went out on my laptop. Once I plugged it back in and booted back up it goes to the Ubuntu logo with the loading dots, but never moves past that point.
How can I get my computer to start again?
Sounds like you'll need to enter recovery mode - as your computer is booting, press the tab key and you should be presented with a list of options to boot from. Select the recovery mode option (this may be in a submenu called "advanced options") and wait to be presented with a menu. There should be an option to resolve package issues (I can't remember the exact title, but it will be obvious which it is). Try running this to fix the problem.