system installation - Can a full version of Ubuntu be Installed on a USB Drive?

I have installed Ubuntu several times on my thumb drive. First times were on 8 GB thumb drives.
I noticed I could not save anything as when I rebooted, it was all gone.

So, I got a 30 GB drive, tried installing again... and same thing.

Now I've found out I am running a LIVE version... not actually an installation.

So, for the somewhat novice who cannot do command-line installs,
step by step is there a way to install ubuntu-12.04.3-desktop-i386.iso using, unetbootin-windows-585.exe to a 30 GB USB stick using my Windows 7 machine?