Android isn't the same on all devices. These instructions are for devices running Android 7.1.1 and up. For most apps, you can control notifications from the Settings menu in the app itself. You can also use your device's settings to completely block notifications from a particular app:
  • Open your device's Settings app [gear wheel icon]
  • Tap Notifications
  • Tap an app
  • Tap 'Block All' or 'Show Silently'
Now you will never get notifications for that app. You can always turn notifications back on if you change your mind. Tip: In Android 7.0 and up, when you get a notification from an app, you can slide the notification slightly to the left or right and tap Settings [gear wheel icon].

For devices with Android 5.0 and below:
  • Open your device's settings
  • Go to the app manager
  • Go to the app whose notifications you want to turn off
  • On the 'App Info' screen you can turn off the notifications for this app
For a device-specific explanation, please visit the Google Support page: