Yes you can install Ubuntu on a USB, but it won't transfer nicely to other systems. I'm running a Lubuntu install from a usb stick right now. Just boot up a installation disc and choose install, then when it comes time to choose the install type pick "something else" and specify that your install be done to the usb stick (eg "/" on "sdb1" use as "ext4"), make sure to specify the bootloader to it as well (eg "sdb"). Ubuntu doesn't care if it's a flash drive, ssd, or hard drive.
To find out the correct device name for you USB drive, you can open a terminal and run
That will list all attached drives, one of which (usually
/dev/sdaX) is your internal hard drive and one (usually
/dev/sdb1) will be the pendrive.
Official Ubuntu Installation Instructions (These are for 13.10, but it's the same):