Ok if you havent checked out the link Uri posted, "root," to put it simply, is administrative privileges on the phone and gives you the ability to mess with core system files and configurations.
The "Superuser" app is just an app that is designed to grant or block root permissions to different apps.
Rooting android phones is sometimes complicated but it mainly depends on the phone and manufacturer as different manufacturers lock down their phones in different ways. Some developers have brought it down to you just plugging in your phone when a certain app is running and everything is done for you while other times will require you to run terminal commands, and download specific software onto your phone. Unlike standard desktop Linux, root has to be acquired through exploits apparent in the phones.
Now why gain root?
Well for android phones, it allows us to a variety of things such as have the ability to underclock our processors for considerably better battery life or overclock them for performance. We can also create wifi-hotspots with ease, run network based attacks such as ARP-Poisoning, conduct cookie hijacking and so on. Its like jailbreaking an iPhone (except its a LOT more power to tinker in the palm of your hands).
In your case, your app probably configures the network card on your phone for the File Server in a way that cannot be done with the traditional Android API and so needs more system access (hence the request for root permissions).
There is no "simple" setting to turn on root no matter what phone you have. This is because carriers make money by locking down your phone and forcing you to pay for more functionality (i.e tethering/wifi-hotspot). If they give you the ability to activate root, then some people could easily circumvent their control, thus making them lose money. The other possibility is that careless people who don't know what they are doing could seriously mess up their own phones and thus making it a support nightmare for carriers and manufacturers who now have to deal with bricked phones.
What should your best approach be? It all depends. If you really plan on tinkering with the phone or want custom software or want better battery life, then rooting is for you. If you just want to use your phone with no extra functionality or no hassle, then don't bother for root as you probably don't need it.
If you do want to root, CAREFULLY follow all instructions and do your research. Your best place to start is on your device forums on xda here: http://forum.xda-developers.com/forumdisplay.php?f=1404
The 1 click root also has a very good chance of working but this isn't verified to work with your device afaik.