It was an AirportExtreme and there was a backup hard drive connected
Good info and a strong clue.
Do you recall whether the hard drive that was connected to the AirPort Extreme was used for Time Machine backups, or other data storage?
If the hard drive was used for Time Machine backups, then Time Machine was programmed to connect to the hard drive for backups. Since the hard drive is no longer there, Time Machine cannot find the drive.
Or if the hard drive was used to store media files......music, video, images, etc......an application (iPhoto, Photos, iTunes, Music, etc) might have been set up to automatically log into the hard drive to use the media libraries stored there If that was the case, then the application is still looking for the hard drive but cannot find it because it is no longer there.
Regarding your penultimate comment:
a) There is one server address listed in the Server Address window (I've no idea what should be there if anything). Would that be my external backup drive?
Possibly, if your external drive is a network drive that connects to the network using an Ethernet connection. But, if your external drive is connected to your Mac using a USB connection, then that connection would not appear under the list of servers.