Installing & running OS X 10.8 Mountain Lion from an external hard drive onto my Snow Leopard Macbook Pro?
It should be as simple as:

Step Zero: Make sure you have a verified, complete, bootable backup of your Snow Leopard setup, "just in case". (And make sure that drive isn't attached when you are installing Mountain Lion, so you don't accidentally choose it.)


Attach external drive to MacBook Pro. Format as "Mac OS Extended (journaled)". Give it a clear and distinct name such as "EXTERNAL DRIVE FOR LION"
Launch App Store.app. Purchase Mountain Lion.
Wait wile the installer downloads. It's ~4gb, so you'll be waiting awhile.
When the installer finishes downloading, it will automatically launch and prompt you to start the upgrade process. DON'T PANIC. Nothing will be done until you tell it to proceed. 
(OPTIONAL BUT HIGHLY RECOMMENDED STEP) Before you start the installation process, quit the installer. Then, go to Finder and open the /Applications/ folder and look for an app called something like "Install OS X Mountain Lion.app". Drag it from /Applications/ to somewhere else, like your Desktop. (You may be prompted to enter your password to do this. That's OK.) Why do this? Because if you leave the installer in /Applications/, it will be automatically deleted when the installation completes. I recommend keeping it around because it is useful if you want to make a USB installer or something like that. Once you have moved it, launch the "Install OS X Mountain Lion.app" again.
Very early on in the process you should be able to choose which drive you want to install Mountain Lion to. (You may have to click on a 'Customize' button in the installer.)
Choose the "EXTERNAL DRIVE FOR LION" drive and follow the rest of the prompts in the OS X installer.
After the installation starts, it will reboot your computer from the external drive and continue the installation. Make sure this is not interrupted.
When it is done, you can choose the default Startup Disk in System Preferences, or hold down the Option/Alt key when the Mac is booting up.


Bonus Tip: QuickBoot is a free app which will let you reboot your Mac using a different drive than the default startup disk without changing the default startup disk.

Just go slow, watch what you are doing, and it should be very straight-forward.

(Nevertheless, don't forget Step Zero. Have you booted from your backup drive lately to make sure it really boots?)