Mail has to be running in order to see notifications, which is an irritation and it would be much better if you were always notified, whether it was running or not. Choose either Banners (these disappear by themselves after a few seconds on the screen) or Alerts (you have to click a button to close the pop-up). Select the Mail app on the left and tick all the options on the right. Go to the Apple menu, System Preferences, Notifications. The idea is straightforward and when a new email message arrives, it should display in a pop-up box in the top right corner of the screen. I found a solution, but you won’t like it. I spent another couple of hours trying to get to the bottom of Mail notifications the other day. It is a nice idea and I love notifications on phones and tablets, but they just don’t seem to work anywhere near as well on my Mac. I mostly ignore Notification Center in OS X because it doesn’t do anything useful. This problem has existed on my Mac since, well, for as long as notifications have been around. Unfortunately, this does not happen for some people with the Mail app on the Apple Mac.
One is that the event should be added to Notification Center and the other is a pop-up notification in the top right corner of the screen. Two things should happen when certain events occur, such as a calendar appointment, instant message or a new email.