Moto Display on Moto X 2014 (on the original Moto X is called Active Display), shows you your 3 last notifications without turning on the whole screen and it lets you interact with the notifications as well. To disable / enable Moto Display on Moto X 2014, do as follows:
1. Open the Moto application on your Moto X (make sure it is updated to the latest version)
2. Tap the Settings button on the top right and then head to the Display settings
3. To enable Moto Display on Moto X (2nd gen), make sure the Moto Display option is marked (Unmark the option to disable Moto Display)
4. To customize which notifications should be shown while the screen is off (using the Moto Display feature of course), tap the Select Apps option and mark/unmark the applications you want to send/not send notifications to the Moto Display)