Backing up an Android device is a broad term, that entails various levels of backup that cover multiple aspects of the data on your Android device, using different tools.
In this post I write about using the Titanium Backup app (or “TB”), which is available to download from Google Play store for free. It also has a paid pro version. TB requires a rooted device to work properly.
TB allows you to perform a user-level apps & data backup – including all (user/system) apps and their data, but excluding low-level stuff like the kernel and the Android stack. This is very useful when changing ROMs and wiping the device data in the process, as it allows restoring the apps on the new ROM including the state and data related to each app.