How to set app notifications as priority on Android 6.0 Marshmallow (stock Do Not Disturb mode)
To set app notifications as priority, open your Android device's main Settings menu, go to Sound & notifications, and then to App notifications. There, you'll discover a list containing all the apps installed on the device. Tap on the app that interests you, and you'll be able to see a "Treat as priority" toggle - which obviously needs to be switched on. When the toggle is on, notifications from that particular app will be enabled during Priority mode.
In case it wasn't already clear, this tutorial was done using stock Android 6.0 Marshmallow. The procedure is similar on stock Android Lollipop, but things may look different on devices running non-stock Android (either Lollipop, or Marshmallow).