Rooting gives you root access. Much like jailbreaking an iOS device. With root access you can overclock your device which speeds it up. Also, you have a great control and access for customization with themes, tweaks, etc. And you can select custom ROMs, which are custom builds of the Android OS. These ROMs have eliminated all the boatware and stuff that slow down your device as well as adding some extremely cool tweaks. I have about ten ROMs saved to my SD card and switch regularly. Personally, I would not have an unrooted Android.
So basically it means giving yourself administrative access. Sounds like something I'll probably do (eventually . . .) Thanks for the info. Since Android is a Linux distro, doing this would be meaningful.
I believe XDA has a toolkit that makes it easier to root, etc. Not as easy as a jailbreak, but it helps for those of us not so confident in our rooting abilities.
I read through it - Also there's detailed explanations and instructions at Phandroid (which seem to be "borrowed" from xda). It looks very easy, except it's a little long-winded. Right now I'm so new at Android (my S3 is my first experience with Android) that rooting is unnecessary, but like I said I may do it down the road as I get used to what this phone can do.