You have to install Ubuntu in your external USB disk. You might need two usb stick or one live CD and a disk will do the job. Please remove any other media connected to USB (Not necessary but it will make the job lot easier).
Now boot from the livd CD or USB and install Ubuntu in the USB disk. Recommended size for the USB is 8GB for fully functional operation. You have to choose something else from the choice menu while installing.
Your primary hard drive is sda and anything connected will be in the same format as 'sdx'(where 'x' stands for b,c,d, depending on your number of disks connected) .Format the system using 'Ext4 File System' and give mount point as '/' this is your root partition. Then you can see a tick near your selected drive 'sdx'. Most importantly there is a option which will ask where the bootloader to be installed you must change it to '\dev\sdb'. Now you are ready just click install. You have to answer few questions like name, location, password etc and you get your working Ubuntu on Usb stick.
You have to tell the boot loader to boot from USB drive each time you want to go to Ubuntu on your external drive.