I have seen the term 'Toast' used in the Windows 8 context. I have also seen them referred to as 'Toast Notifications'.
What exactly are 'Toasts' in Windows 8?
Windows – What are ‘Toasts’ or ‘Toast Notifications’ in Windows 8
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Related Solutions
Just go to Settings charm / Change PC settings / Search & apps / Notifications and enable toast notifications for the relevant apps (toggle the setting if already enabled):
(This answer is more of a summary of the work done in the comments of the post.)
Notifications are lousily done in Windows 8.1, but I advised to still recheck all settings and include their screenshots in the post.
As the notifications you were missing the most were from the email app, you checked it and found that the app was set to show notifications for emails from "favorite persons", which you then reset to show notifications on all incoming emails. Your explanation was that this change was caused by the Windows 8.1 Update 1, and the resetting of this parameter fixed the notifications problem for emails.
In the settings for the Facebook app you turned on all notifications (even though this is a bit of an overkill), but this didn't help. As this app is notoriously buggy, this was taken to be a bug that might be fixed in a future version.
My final advice was to keep the Facebook app updated and recheck its settings after every update, to see if some new setting has now become relevant. Hopefully this will be fixed in the future.
Best Answer
Windows 8 has a notification system similar to that implemented in mobile operation systems such as Android, IOS and Windows Phone.
A Windows 8 Toast is a notification triggered by an application or the operating system itself that is displayed to thew user by way of a small pop-up notification.
Wikipedia defines a Toast in the computing context as:
Toasts in Windows 8 comprise of a short text message and can optionally include a small image.
More technical info here.