Flappy Bird became the most popular game in the world in 2014 for a brief period. However, the application was taken off the app stores by the Vietnamese developer Dong Nguyen due to person issues.
Dong is the creator of Flappy Bird, thus taking the game down from the app store was ultimately his decision. You are no longer able to download the game to play from the App Store or Google Play.
The easiest way to have this game on your iPhone or android device is to download the IPA or APK files, then copy the file into your devices. For the game to work on iOS devices, they must be jail-broken first.