Firstly, a root is not only a part of a tree but also means the traditional name for the account in the unix system which gives the user the power to fully control the system (it is a sort of administrative account). Android's operating system is based on the Linux core and also has the user named root which has the unlimited access to all files and can perform all of the actions concerning their modifactions.
Root, in the simpliest definition, is the access to all of the Android's system files and the the possibility to modify them. There is an application - SuperUser/SuperSU - which is used for managing those actions, as well as CWM (ClockWorkMod) and TWRP (Team Win Recovery Project) which are more advanced Recovery Modes than those in a standard Android devices.
- It enables any system modifications
- It enables uploading custom ROM
- It allows you to make a full backup of the system and the application
- It enables you to install applications on a memory card
- It allows you to uninstall any unwated apps (e.g. those directly from the producer)
- It enables you to change the initial screen.
- It may lead to the instability of the system
- By rooting, you are installing an unofficial operating system which will lead to loss of warranty.
- It attracts more viruses