New Android Version: KitKat

Sure! So rooting basically gives you "administrator" privileges over your android phone (if you're ever heard of people "jailbreaking" their iPhones, it's basically the same thing). When you root your android phone and gain administrator privileges, you can download (known as "flashing") many different ROMs, which are basically modified versions of Android.

The main benefits to doing this are to customize the hell out of your phone, gain access to a lot of really cool apps in the play store that you would otherwise be unable to download (like apps that let you download paid apps, or apps that get rid of ads on ad-heavy apps), be able to actually back up your entire phone (apps, user data, files, etc) and enhance certain aspects of your phone like wireless WiFi tethering (letting your laptop grab WiFi from your phone without paying anything extra, wirelessly).

Many different groups of developers develop different types of ROMs. For instance, if I have a Galaxy s4 and hate the user interface, I can root my phone and then download a ROM that completely mimics the user interface of an iPhone, or a theme that enhances certain features that are important to me. Since the s4 comes with a lot of "bloatware" (software pre-installed by Galaxy that bloats the phone with unnecessary crap), rooting an S4 and flashing a custom ROM typically speeds up the phone immensely. There are lots of ROMs out there, and you can switch between them at any time provided you back up all your apps and data beforehand. Note that flashing a ROM will moot any Android updates, but that's basically the reason why people flash ROMs - to get enhanced features not found on Android. At the same time, while stock Android updates usually take a long time to be baselined for users to actually download, updates to ROMs are typically way faster. I currently have a ROM called Paranoid Android, and updates to that come up literally once a week - and every few updates there's always a sweet new feature added, like allowing me to go into "Immersive" mode, which basically hides the top and bottom menu bars and full-screens your phone.

Kernels are another thing to mention - flashing a kernel is typically associated with vastly improving battery life, enhancing speed, etc. I've never done that before, though, and have very limited knowledge of all the benefits of flashing a kernel.

I don't have much time now to look for links, but if you guys want, I can post a few a bit later. There is a tiny learning curve when doing this if you are interested, but honestly it doesn't take that long.