How do I know if my ArcGIS 9.3.1 has a GIS server on my computer ?
A quick way to tell if an ArcGIS Server is up and running on your computer is to check via the Rest API:
If you get to a page with some basic map service related options, the service is running. If you get an error, ArcGIS Server may be installed, but it's probably not running.
EDIT: Response to comment
Here are the default URLS for the REST API for both editions of ArcGIS Server:
These are the default URLs unless changes are made to the configuration of the REST API. If you've gotten that far, you probably know whether or not ArcGIS Server is installed on your computer.