Before we go any further I think that it's important that I point out the dangers associated with this process. Specifically, if you make a mistake and install Ubuntu onto your system's hard drive then not only have you lost data, but you've become a Linux user. You need to make sure that you choose the right drive.
Tip: If you want to eliminate the chance of overwriting one of your hard drives then you can physically disconnect them before booting from the Live CD.
Note: Back up the data on your USB flash drive if you don't want to lose it!
Here's the process:
Boot up off the Ubuntu Live CD.
Kick off the Ubuntu install process.
Word your way through the guided install until you get to the Step 4 - Prepare disk space. Here make sure to choose Guided - use entire disk.
Note: I've only got one drive displayed - all others have been disconnected.
Carry on working your way through the install until you get to Step 7 - Ready to install. Here click on the Advanced ... button. Under Advanced Options make sure that Install boot loader is checked and from the Device for boot loader installation drop down box make sure to choose the correct drive to install the boot loader onto - this is of critical importance if you still have your hard drives fitted! When you're done click OK and continue with the install.
Now it's mostly a matter of playing the waiting game as Ubuntu is being installed on your USB flash drive.
When the process is done, remove the Live CD and reboot.
Note: If you disconnected your hard drives, now is a good time to reconnect them!
Remember that to make use of your Ubuntu-powered USB flash drive on a PC you'll need to use the BIOS or boot loader to choose the option to boot from USB devices.
Note: If you have VMware Workstation installed then you can create your Ubuntu-powered USB flash drive from the comfort and safety of a virtual PC - also, since you can boot up off the Ubuntu .ISO file, you don't need to burn a CD!