Google Android is/was based on Linux, it has a Linux kernel at it's core.
On a Linux OS, a person using the super user account had complete control of every system function, there was nothing barred or off limits. The Super User account could be used to destroy the entire system once. The idea of having superusers in Linux was that anybody even administrators could use a normal account, If and when they needed access to something, assuming they knew the Super User password, they could log on to the super user account and perform the restricted option, or on newer Linux versions give permission for that singular activity, or run an su command. This extra level of security protected the system from malicious code, viruses and even unwitting administrators. If people only ever used the super user account when they needed to and a normal account for normal use malicious code would not be able to destroy the system without root access.
So this is the brief history root access. Anyway all of what semperLux said is true, root access on an android device can also give you access to the terminal, a command line interface. It allows you to install certain apps which are unavailable with our root access, such as apps2sd.
Many manufacturers limit root access because they don't want to sort out ruined phones because people didn't know what they were doing and deleted files by accident. Many operators don't want it because it can be used to get rid of their restricted (uninstallable apps) and enable things like wifi tethering without a tethering agreement.
Therefore please note that if you root, it will often invalidate the manufacturer's warranty.