A Today Extension is a widget shown in the today view of the notification center of OS X and iOS.
Usually a today extension shows informations which is related to current events, like current sock prices, weather and upcoming events. Some extensions though provide other functionality like a calculator or iTunes controls.
Today Extensions were introduced in iOS 8.0 and OS X 10.10 and can be published by developers through the App Store.
App extensions in the Today view are called widgets. Widgets give users quick access to information that’s important right now. For example, users open the Today view to check current stock prices or weather conditions, see today’s schedule, or perform a quick task such as marking an item as done. Users tend to open the Today view frequently, and they expect the information they’re interested in to be instantly available.
To read more follow this link - App Extension programming guide