Let's say your system just melted down after an upgrade, or your new kernel won't boot. You can't fix the problem with apt-get, because you can't even get to a command line; the kernel just spews out errors and hangs on bootup. Thankfully, with a live cd, you can repair your system and get it up and running. You have 2 options for the live cd: Knoppix or the Ubuntu live cd. Since Knoppix generally has better hardware detection, this will be used as an example.
First, download the iso from http://www.knoppix.org/
and burn it to a disk.
Get your BIOS set up to boot from the cd, pop in the Knoppix disk, and boot.
Your hard drive should show up on the KDE desktop as hda1 or sdb2 or something, depending on your system.
Click on it to mount it, then right-click, actions -> change to read-write mode. It'll pop open a dialog; click yes.
Now, open a root terminal, found in the Knoppix menu (the one next to the K on the panel).
Enter: chroot /mnt/hda1 or whatever the icon for your harddrive says on the desktop.
You can now use all the commands on the hard drive, including apt-get.
If you ever get this error: /dev/null: Permission denied, do this: rm /dev/null and it should go away.
Now, use apt-get to upgrade your kernel (apt-get install linux-386) (or whatever processor you're using), udev (apt-get install udev), or anything else that's messing up your system.
Reboot, and enjoy your newly rescued system!