Notifications in OS X can be a bit complicated, and the User Interface might have been better. Basically, there are notifications themselves as well as the OS X Notification Center where they are listed. This how-to shows the basics of managing both.
The place to start with managing OS X notifications is System Preferences > Notifications. It will look something like this.
Fig 1. System Preferences > Notifications
The first thing to notice is that there is a distinction between a notification itself, call it an event if you will, and an app's participation in the Notification Center where notifications are listed.
The Notification Center is displayed by clicking its icon at the extreme right of the menu bar.
In OS X Mavericks, the entire display slides to the left. (In OS X Yosemite, it will simply overlay.) Notification Center looks like this:
Fig. 3. Notification Center. Image credit: Apple.
The Contents of Notification Center, when displayed, is dictated by the sections in the preferences (Fig 1.) called "In Notification Center" (INC) and "Not in Notification Center (NINC)
1. Moving an app in/out of Notification Center. In the list on the left side of Notification Preferences (Fig 1.), drag an app from INC to the section below, NINC.
This manages the contents of Notification Center (Fig 3), but does not dictate whether individual notifications will appear as banners or alerts at the top right of your display when they're generated.
I experienced a problem dragging an app from INC to NINC when NINC is initially empty. It may be a bug or some other issue. I solved it by seeding the NINC with the checkbox "Show in Notification Center" to the right of the app selected. See Fig 4 below. After that, I had no trouble dragging an app from INC to NINC.
Fig 4. App options are in checkboxes
2. Manage individual notifications. After you set the INC and NINC lists, you can configure individual notifications. Select one, and you'll see something like Fig 4 above. Different apps may have a slightly different set of checkboxes depending on how they operate. If you don't want individual app notifications to pop up in the display at all, change the alert style to "None".
Fig 5. Alert style.
3. Suppress All Notifications. If you scroll to the top of the INC list, you'll see a lunar icon with the label "Do Not Disturb." This was added in Mavericks, and allows you to set a time window during which all notifications will be blocked from appearing on your display. See Fig. 1. For example, I have notifications blocked during the day when I'm writing or podcasting but the block ends in late afternoon when I tend to do housekeeping on my Mac.
4. Reference. For a complete table reference on Notification Preferences by Apple, see Knowledge Base article #PH14360.
Once you distinguish between notifications appearing in the Notification Center and the control of how individual app notifications are announced, things get a lot more understandable.