Apple - What are the blue dots next to App names in iOS and macOS?

The blue dot is the replacement to the banner that iOS 6 had for a new app.

There are two conditions for a banner being shown:

• an app has been downloaded from the App Store but has not been opened yet

  (a restore from iTunes/iCloud will also cause an app to be classed as 'New')

• an app has been updated but has not been opened since it has been updated

  (this counts for manually updated apps or apps that have been automatically updated by iOS 7's new background app updater).

As soon as the app is opened, the dot is removed.