As far as computers are concerned, there are generally two fronts. The first is the ability to run your lights entirely from a software-based controller (like CompuWare), connected to a USB widget/dongle to send the DMX signal to your fixtures.
The second one is the ability to trigger your regular DMX controller/console trough the use of MIDI triggers from a computer. In this situation, everything is programmed into the controller just like you would do for normal operation via the buttons/faders on the actual controller. Then, you send MIDI commands from a computer through the MIDI input jack in the back of the controller (assuming it is a MIDI-enabled controller in the first place) to trigger the appropriate scenes, chases, show steps, etc.
In addition, some controllers have an RS232 jack on the back, or even a connection that would go to a computer monitor. All in all, these functions are limited in their abilities (at least in DMX consoles in this neck of the woods) . An RS232 connection is useful for connecting to a computer to update a controller's internal software/firmware or even to dump a memory file for storage.
If you have a regular controller like the DMX Operator 192 (or the Operator Pro), Show Designer, etc., you are not going to be able to hook the controller into the computer and program and run your lights exactly the same way as just working on the physical controller itself.
Like I said, I think what you're getting into is the use of MIDI, which requires pre-programming on the controller and triggering those scenes, chases, and shows from a computer.