If your bios allows you to boot from the external drive, you can put
Grub on the external drive. If you put Grub on your C: drive and take
the external drive away, then the system may not boot.

If you put Grub on the C: drive and want to get rid of it, you will
need to re-write the Master Boot Record. In Vista you would use your
install disk, go to repair an installation, open a command window, and
then enter bootrec.exe

If you install Grub on the external drive, you will probably need to
edit it to get it to boot properly.

When you boot the first time, it will likely fail with error 17.
(Linux is not where it wants it to be.) During the installation, the
external drive is seen as drive 1. During booting from the external
drive, it becomes drive 0.

So you need to edit the line that says.

boot (hd1,0) to boot (hd0,0)

You can do this by following the prompts in Grub.

To make the change permanent. Once you are in Linux, open a terminal
window and enter the command:

pseudo gedit /boot/grub/menu.lst

This will bring up the file, and allow you to save it to the drive
when you are finished.

As for taking the drive to another computer. I have not tried that.
I have Linux working on an external USB hard drive, but I've not
tested it on different machines. I would imagine it would depend on
how similar the machines were.