For devices with Android 6.0 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 app to completely block notifications from a particular app:
  1. Open your device's Settings.
  2. Go to Sound & notification > App notifications.
  3. Go to the app you want to turn off the notifications.
  4. Switch on Block all.
Now you will never get notifications for that app. You can always turn notifications back on if you change your mind.

For devices with Android 5.0 and below:
  1. Open your device's Settings.
  2. Go to the app manager.
  3. Go to the app you want to turn off the notifications.
  4. On the 'App Info' screen you can turn off the notifications for this app.
For a device specific explanation please visit the Google Support: