You can keep your Lightroom catalog and images both on an external drive.
It depends upon you while importing an image from an external drive, you want to be on your computer as well.
You can just ADD images from external drive while importing which will not import images to your desktop and just refer to the external drive image.
Cool! I'm glad to hear it's possible. Could you perhaps explain how you can transfer all of the lightroom information to me external hard drive? Thanks again!
You can also simply move your catalog file at any time. Just make sure Lightroom knows that you have made this move by double-clicking it once you've moved it. This will open the catalog from the new location, alerting Lightroom to this new location. See: LR3 – Where is my Lightroom Catalog on my Hard Drive? « Julieanne Kost's Blog