Unlike modern games downloaded directly from the Play Store, playing Quake 3 on Android requires a "Source Port." Since id Software open-sourced the game engine years ago, developers have created versions that run on Android.
You need two things:
Price: Paid (~$5.00 on Google Play)
Best for: Users who want a one-click solution with premium support.
Quad Touch is not free, but it is the gold standard. It bundles optimized OpenGL ES rendering, full mod support, online multiplayer, and a highly customizable touch overlay.
The most stable, feature-complete port is QIII4A (Quake III for Android). It is based on the ioquake3 project and supports high-resolution textures, touch controls, external gamepads, and online multiplayer against PC players via standard Quake 3 servers.