If you have both your images and your catalog on the same external drive, you simply find the file 'catalog.pseXXdb' in the HDD. That main sqlite database will be inside the catalog folder.
Once the HDD is connected to another computer with the same pse version, just double click on the catalog.pseXXdb to open the organizer with that catalog. The location of the catalog will be remembered by the new computer.
1 - catalog.pse8db is not the catalog, it's the main part of the catalog, which is the whole containing folder
2- Having both the catalog folder and the pictures on the same HDD generally means that you have restored a full backup to the HDD
3 - Keeping both the catalog and the files library in the HDD is the best way to share your work between two computers by just plugging them alternatively.
If you have Elements 8 working on your computer, go to File>Manage Catalogs and navigate to the folder that contains the catalog.pse8db file. You should then be able to open the catalog. Otherwise, double-click on the db file itself and it should open automatically in Elements 8. (I no longer have that version on my computer, so can't test whether this will work, but it should.)