There are several great projects you can do with an Arduino, and if you're just getting started, you can use other people's code. However, as the video above shows, you can also create your own simple apps for your Arduino project using MIT App Inventor.
Dejan Nedelkovski uses two examples: controlling a simple LED and controlling a Stepper Motor. In the 9-minute tutorial, you'll see how to connect the MIT App Inventor with your phone and add the needed functions. The tutorial runs a bit fast, and you'll probably want to check out Nedelkovski's previous tutorials if you're not sure how to connect your phone to the Arduino board over Bluetooth.
Still, a nice beginner video to watch in honour of Arduino Day (today!).