One more thing...when I try to launch from the external drive on to the desktop, I get this error message: LIGHTROOM cannot launch with this catalog. It is either on a network volume or on a volume on which Lightroom cannot save changes." So I don't know what to do. Any help would be greatly appreciated.
Having the catalog and your images on an external hard drive is the logical and best choice. You should be able to simply use Lightroom to open the catalog on that external hard drive and work on either computer. You might want to check your read/write permissions for the drive to make sure you have both on both computers.
You don't need LR on your other Mac to backup your images. LR is a database program. Your images are not IN LR. LR only References, creates a record, of you images in the database file, IE the LR catalog file.
You backup your images to another external drive by Copying them with Finder. By default LR backs up the catalog file once a week and it stores the backups in the same folder the catalog file is. I suggest you move the LR catalog backup location to one of your external drives. You do that when LR Prompts you to make a backup. In that dialog box you can select a location to store the backup files.
And Yes you can install LR on your other Mac and then copy over the catalog file and connect your external that holds your images to that other Mac.