Karen,
You probably have it set up correctly. You started the ArcGIS Administrator which is a portal you used to point to the ArcGIS License Server Administrator (hosting the licenses). When you start the ArcGIS Administrator, it will always display the hostname of the machine it's installed on. The hostname in the upper left hand is your machine's hostname. If you want to verify, just open a command prompt and type in hostname. It will display the hostname of your machine. The one in the middle is the hostname of the machine hosting your license manager. If that is the correct license manager hostname, you can go to the Availability folder and display all ArcGIS Desktop licenses hosted on that license manager server. Therefore the names you've circled in red should always be different unless you have the license manager installed on the same machine. In this case, you're pointing to your own machine for the licenses.
If there are licenses available, you should be able to start ArcMap or ArcCatalog.
Hope that clears up any confusion.