The function of a drive inside or outside of your computer is very similar.
Most external drives are very similar to internal drives, except the interface used to connect them is usually USB or Firewire. There are external cases you can purchase that accept IDE and SATA drives.
As far as running the drives, you can run them much like you would your PC. For many USB externals, the power is delivered via the USB port, so when the computer shuts down, so does your drive.
I personally keep my computer on most of the time, and have both dedicated externals and internals as well as converted internals. Despite wanting to leave the computer on, I do shut it down at least once a week.
Hope this helps!
It is still better to shutdown or turn off your external hard disk drive if you are not using. Remember that using your external storage device frequently and in an extended time would significantly reduce the lifespan of the hardware device.