External hard drives are a great way to back up all of your information, but they can also serve as storage for files that are taking up room on your computer's hard drive, including music and other media. While you can't move iTunes itself to your external hard drive, you can use it to store all of your media files.
iTunes on Your Hard Drive
No, you cannot install iTunes onto your external hard drive. You can put the files for the program there, but the program would still install, and run on the computer to which the hard drive is connected.
Music on Your Hard Drive
If you are looking to save space on your computer's hard drive, you can put your music on your external hard drive. iTunes can be set up to play music from any source, including external ones.
iTunes is set up to play music from a very specific location, and moving those files will confuse the program, and leave it unable to play any files. When you attempt to play a file iTunes cannot find, it will not play, and a small exclamation mark will appear next to the title. If the files are moved, their location must be changed in the program's preferences as well.
Moving your Files
If you wish to move your files to an external hard drive, connect it to your computer, then copy the contents of the folder, "iTunes media library," onto the drive. This folder contains all of your iTunes media. Then, go to iTunes' advanced preferences, and change the "iTunes media folder location." You can browse your hard drive, and find your files there. Once this is complete, iTunes will play your music from your external hard drive, and you can delete the folder that remains on your computer's hard drive. But remember, the external hard drive must be connected for you to access your music.