If you aware in Spring '14, admins can now provide more detail within the notifications including the text of a post or comment in Chatter, a Chatter group’s name, or a task description. Salesforce1 notifications let users know when certain events occur in Salesforce, such as when they receive approval requests or are mentioned.
Two types of notifications can appear to Salesforce1 users:
1. Push notifications are alerts that appear on your mobile device when you're not using the Salesforce1 downloadable app. Push notifications are not supported in the Salesforce1 mobile browser app. These alerts can consist of text, icons, and sounds, depending on your device type. You can choose whether to receive push notifications; however, you can't enable push notifications if they were disabled by your administrator.
2. In-app notifications keep users aware of relevant activity while they’re using Salesforce1. Users can access their 20 most recent notifications by tapping the Notification icon.
Any time that you @mention somebody in the Chatter feed they will receive both a push notification and notification within Salesforce1 app.
All you need to do to start using these notification is navigate from Setup, click Setup > Mobile Administration > Notifications > Settings, and then select both Enable in-app notifications and Enable push notification. And that’s it!
Some notifications include text that your users enter in Salesforce. To ensure that sensitive information isn’t distributed through a third-party service, by default, push notifications to iOS devices include the user’s name only.
For example, an in-app notification might be “Allison Wheeler mentioned you: @John Smith, can you update the Acme account with their new shipping address?” The equivalent push notification for iOS devices would be “Allison Wheeler mentioned you.”
To control this, go to Setup > Mobile Administration > Notifications > Settings and Include full content in push notifications.
Screenshot below show when "Include full content in push notifications" is disable and in phone setup fo Notification Centre is set to Alert.
If you enable Include full content in push notifications,it will show some of the message in push notification. Screenshot below taken when phone is locked and "Show on Lock Screen" enabled.