Added on: 14-12-2012
Last update: 18-07-2015
Tor is both software and an open network that helps you defend against network surveillance that threatens personal freedom and privacy, confidential business activities and relationships.
Orbot allows access to Tor by accessing a local SOCKS or HTTP proxy. On a rooted device, the proxying can be completely transparent i.e. the app that accesses the network need not be aware of the proxy's existence; you can choose which apps go via the proxy in the settings.
If you don't have root access, there are some apps that are designed to work closely with tor or allow proxied connections: ChatSecure, Lightning and Twidere. There is also a proxy configurator addon for Firefox called ProxyMob (not yet available from the Mozilla addon site).
This app will use root access if you have it, but it is not required. Root access is used to transparently proxy apps, which allows you to force any app to use Tor.
Currently there are two build falvors available, PIE and noPIE. "Position independent executable" is a technique that got introduced somewhere in the Android 4.x cycle and the PIE builds should be default for Android 5+. For technical details see Wikipedia.
Basically you should use PIE builds when they are available for your device.