One of the amazing things you can do with United when booking a roundtrip award ticket is a concept called a stopover. A stopover is different from a layover in that a layover allows you to stay in one city for < 24 hours. In a stop over situation, you can stay in a transit or connection city for > 24 hours.
I use United as an example, because their frequent flying program offers 1 stop over on round trip international tickets. You can further come up an awesome itinerary by leveraging open jaws which means you are allowed to leave from another airport than the one you arrived to.
Back to stopovers. United rules how limits stop overs only on international round trip tickets. So no one ways, no domestic stop overs.
An example of a stop over,
To: New York to Hong Kong (stopover) to Bangkok (final destination)
Return: Bangkok – Tokyo (layover) – New York
So in this scenario, you could theoretically stay > 24 hours, a week, or several months in Hong Kong as you continue on to Bangkok.
Thus, you have a free trip to discover both Hong Kong and Bangkok.
With United, you can stop in any city that United or its partners on Star Alliance flies to.
You can also do the stop over in any city reasonably in your route. This means, if you were to fly from New York to London (stop) to Bangkok, they may not allow you back track.
If you want to get really creative, you can visit 3 cities using a Open Jaw. I will explain more about about an Open Jaw ticket in the final series.