Extentions let you add apps to the Share Menu, Today in Notification Center, and Actions. For instance with the Share Menu you might add Facebook and/or Twitter so when you select a file you can click the share button and upload it. Or for Actions you might enable the whole Adobe suite so when you are editing a app in photos you can send it to any of the Adobe applications. With Today you can add a 3rd-party extention to the Today view in the Notifications Center for instance you can place the iTunes controls in Today view so you do not hav to keep going back and forth between the app you are using and iTunes.