What is a "round-robin flight", in layman's terms?
It's simply a flight that returns to the original airport after stopping at one or more other airports.  ATC doesn't have a good way to code those types of flight plans in the computer, so the pilot will usually just tell the controller that he/she is planning to land at the intermediate stop for x amount of time.

The controller will then either close their flight following (or IFR clearance as appropriate) and have them reopen when they take off again, or just keep it open.  It depends on how busy their airspace is.  Often they'll just leave it open if it's in the middle of the night.