Apps are the most common type of program on iOS, and the type that users are most familiar with.
Most apps have an icon on the home screen, and display a full-screen window when launched.
iOS includes several built-in apps, such as Notes, Phone, and Weather. The home screen, called SpringBoard, is also an app (albeit a very special one).
Everything that can be installed via the App Store is an app. Apps can also be installed via Cydia, though most of the packages available via Cydia are not apps.