What is root in a mobile and what does it mean?

Rooting means to have administrator rights, gaining root access, or ability to access and change, system root files. This is like accessing and changing Windows operating system files and registry items.  So, some apps (which are a lot), need root access to work. This could be called in other words having administrator rights to use this app. (which sometimes might use a system file, and sometimes even harm your device. So be caucious what app to install).

                 There are far more benefits to root a phone than there are to leave the phone as is. Disabling many of the ads if not almost all ads showed in your apps is a huge benefit, and even saves money over the ad-free version of the same app. You can remove carrier installed apps that otherwise could not be uninstalled, giving more space back on the phone and possibly memory as it won’t run any more. Under-volting and overclocking the processor can improve performance while improve battery life. There are tons more benefits. Plus, if you want to unroot the phone for sending the phone in for repairs is much simpler to do.

Having a rooted device you can install custom rooms, create partitions in your sd card and a lot more. In my opinion, the best thing to do, is to google more info about rooting. I stayed about one week reading only about rooting and its good things!
You can read even this english wikipedia post, if you want:
http://en.wikipedia.org/wiki/Android_rooting