You can use an external HD for backing up your internal HD. This is critical to any type of computer in the event of hardware failure or other type of loss like a natural disaster. In the event your computer loses important information (photos, music, important documents, etc..) you want at least one back up of that important information otherwise you could lose it forever.
I believe when you refer to an "external hard drive that I can run programs from" you may be referring to a bootable clone of your internal HD. This essentially is an exact clone of your computer's internal HD and is a backup! This is extremely useful in the event a computer failure or other type of loss. I your internal HD crashes you can start and run your computer from the bootable clone. This is useful until you can replace the internal HD.
OS X 10.5 & 10.6 have a tool built into it called Time Machine, I will let Apple explain TM. Simply
click here.
Many people (myself included) have redundant backups, this is just in case a backup fails. Personally I use Time Machine and on a separate external HD have a bootable clone. It's wise to backup the backup. Here is a
recent article from MacWorld magazine that talks about backup and one user's strategy.
Roger