You can build many different types of apps with App Inventor. Often people begin by building games like MoleMash or games that let you draw funny pictures on your friend's faces. You can even make use of the phone's sensors to move a ball through a maze based on tilting the phone.
But app building is not limited to simple games. You can also build apps that inform and educate. You can create a quiz app to help you and your classmates study for a test. With Android's text-to-speech capabilities, you can even have the phone ask the questions aloud.