Thanks to iant, dariapra, RK, & rangvald, as the combination of setting up the Geoserver Proxy in the interface, and making changes to the Apache & Tomcat worked. I can now click on the Open Layers and a new window pops up with a blank image within the map surrounding box, all within out IP Address.
So, Solving one issue raised another.
I now need more assitance to fix the new pop up window (after I click the OpenLayers from the Layer Preview pane) in order to show the map that is to be displayed. Within the window, there is the bounding box, text in the bottom left hand corner "Click on the map to get feature info" as well as the bottom righ hand corner "location", and in the IE window, the bottom left identifies a yield sign with paper behind it saying "errors occured".
Is there a way to set this up so the map shows wihtin the new pane after clicking the OpenLayers?
Again, thank you to those who have assited me.
My initial question was: How do I configure Geoserver to show my shapefiles/maps on my Webserver(a specific location as http://www.name.net/geoserver) at work, and not on the localhost:8080? The web server is running Apache and also tried the Tomcat install. Any help is greatly appreciated.