To build an Android app that displays air quality data for a city, you can use the AirVisual API. First, sign up for a RapidAPI account and subscribe to the AirVisual API. Then, create a new project in Android Studio and add the necessary resources for displaying images and text on UI. Next, write the code for the activities that will display the air quality data, including a spinner to select the city and a second activity to display the results. You can use the `GET auto-complete` endpoint to search for the city id and then retrieve the air quality index using the `GET cities/get-measurements` endpoint. Finally, update the manifest configuration to enable internet access and run the app on an Android emulator or real device. The app will display the air quality data for the selected city, including the current measurement, historical data, and density of pollutant components.