The main component of Processing for Android is the Android mode, a programming mode for the Processing Development Environment (PDE) that adds all the options needed to run Processing sketches on Android devices and also in the emulator.
You could also use only the Android core library to develop Android apps with the IDE of your choice, see this tutorial for more details.
The latest stable release of the Android mode can be installed through the Contribution Manager (CM) in the PDE. To open the CM, click the arrow in the mode selector in the menu bar, and choose "Add Mode...":
Once the CM window shows up, select the Android Mode from the list, and then click the install button in the bottom of the window:
After installing the mode, you will be able to switch into it using the mode selector in the menu bar. Follow the Getting Started tutorial for the first steps with the Android mode.
Pre-release versions, as well as older stable releases, are are not available in the CM, and need to be installed manually. In order to do this, first download the AndroidMode-xyz.zip file for the desired release or pre-release from the releases section in the GitHub repo of the Android Mode. Uncompress the zip file and then move the AndroidMode folder into the modes folder inside inside the sketchbook folder.
The 4.0 pre-releases bring several improvements to the Android mode, incuding new functionality for creating live wallpapers, watch faces, and Cardboard apps. You will find more on how to use this functionality in the tutorials and the reference pages of this site. In order to install a 4.0 pre-release manually, go to the GitHub releases section, look for the packages corresponding to Android Mode 0254 (4.0-beta1) or higher, and follow the instructions in the previous paragraph on how to install a pre-release.