Apple's Notification Center in OS X is a relatively new service that offers notifications along the right-hand side of your desktop. This service is used by both third-party applications and Apple programs and services, and offers two configurable notification types: Banners and Alerts.
These two types differ in that alerts will stay present until you interact with one of their buttons, and banners will appear for a short while and then disappear from view. While useful for a quick notice while you are at your system, the default banner dwell time of a few seconds might be either too slow or too fast for your liking.
Luckily you can change this. As with many system services, the Notification Center contains a number of hidden settings that can be activated and configured using the "defaults" preferences manipulation Terminal command. To do this for banners, open Terminal and run the following command:
defaults write com.apple.notificationcenterui bannerTime SECONDS
In this command, replace SECONDS with the number of seconds you would like banners stay until they disappear. When done, log out of your system and then back in, and the new setting should be adopted by Notification Center.
To undo this change, simply remove the "bannerTime" settings entry, by using the following command:
defaults delete com.apple.notificationcenterui bannerTime
Follow this command by logging out and back into your account, and Notification Center should now use its default dwell time.