Solved! Go to Solution.
ArcGIS Server delivers and uses an embedded Java based web server. You know this when you specify port 6080 or (for secure services) port 6443.
ArcGIS Server also offers a web adaptor product which uses IIS which acts as a proxy to the embedded Java based web server(s). By default, the way you access IIS is via port 80 or (for secure services) port 443. If you use these default ports then you do not have to specify them in the URL.
Whether or not you use the web adaptor depends on your configuration. For example, if you have multiple GIS Servers then the web adaptor will distribute the incoming requests to them. It also allows you to rename the instance and provides you with the full functionality of IIS. So, you can either choose to deploy the web adaptor or to expose out the embedded Java based web server.
With regards to which path to use, the answer is that it depends on what you are trying to do.
http://localhost:6080/arcgis/admin/ - Administrative functionality, i.e. how you configure ArcGIS Server itself.
http://localhost/arcgis/webadaptor - Web Adaptor configuration
http://localhost:6080/arcgis/manager - ArcGIS Server Manager
http://localhost:6080/arcgis/rest/services - REST Services, i.e. this is the real functionality that your Flex client will use