Rooting in basic terms is unlocking the phones internal system memory, this gives access to the internal system files of the phone and gives the user the ability to edit those files, install new files, run root only apps, flash custom ROM's, install ClockworkMod and many other features.
Usually in rooting a device your are attacking the device with malware known as an Exploit, the exploit such as "Rage Against The Cage" or "psnueter" use script to attack the system security files until root is gained.
The term root comes from the Linux operating system world, where the most privileged user on the system is called the Root user whereas with Windows that privilege is known as Administrator.
The bad part of Rooting is it generally voids the warranty if your device!
Usually when rooting your device busybox files are added to your phone and the Superuser app is installed, this application protects your device from unwanted access to the system files as you the user must give permissions via Superuser to anything requiring root access.
Busybox is a set of tools for root access, it gives your device LINUX/UNIX based commands for root level tasks.
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