To begin, "rooting" gives you full access to system files and more control over your phone. For instance, you can add custom themes, change your recovery image, alter your bootloader, alter the "toolboxes" so the Linux system operates more efficiently, and add Linux Binaries.
After you root you will be able to put special apps on your phone such as Super User which allows certain apps to access root privileges. Once you have Super User, you can install and use other apps to control the start ups of other apps, limit the running of certain memory draining apps to increase battery life and responsiveness, and install either wired tethering or wifi tethering. Also, finally, you can download an app that reboots your phone for you.
Another great tool through rooting is you can backup your files and ROM onto your SD card.
Also, through rooting you have the opportunity to dramatically change the look and feel of your phone. As mentioned above you can install customized themes onto your phone. Another great advantage is you can install custom made ROMs, kernels, and other files to enhance the way your phone works, looks, and feels.