If you have version 4.5 or higher:
To turn on push notifications, go to the app's menu, tap on settings and tap Push Notifications to On.
To turn off push notifications, go to the app's menu, tap on settings and tap Push Notifications to Off. If the setting has already been set to off, please toggle it back to on and then to off to reset your settings to off again as there was a bug in the most recent version of the app.
For those with previous versions:
From your device (on Android OS version 4.1 and later): Go to “Settings” app > “Application Manager / Apps” > “Dictionary” > tap the “Show Notifications” checkbox
From the push notification itself: Long press on the push notification > tap on “App Info” (launches Dictionary.com app settings) > tap the “Show Notifications” checkbox