While they are trying to be helpful, the answers here assume the OP knows Linux, or what a Super User is, or the difference between having or not having Admin rights in Windows. And yes, people should search first, but you know it will never happen that 100% of the people will do what they should (that's why there are iPhone users. ;-) )
To clarify from above: A phone that is NOT rooted can still be modified (you can still make it LOOK different) by using applications like Launcher Pro, for example, and installing Themes. You don't need to root to do this. If your phone is NOT rooted, you can also still access files that are on the SD card and/or on the internal memory, you just can't access _all_ the files on the internal memory. This is an important distinction imho because a complete newb might misunderstand and think he/she needs to root to access
any of his/her files.
1. If you Root your phone, you will be able to access files that you are not allowed to access without root. These are usually important system files, which you don't want to touch unless you know what you are doing because you could screw up your phone.
2. If you root your phone you can use certain apps that need to access parts of the Android system that can't be accessed without root, but these are mostly obscure (in the past you had to root to tether your phone to your computer to share your data for internet access, for example, but that is no longer necessary).
3. Rooting lets you delete the bloatware/crapware that often now comes pre-installed by the manufacturers and carriers, and which usually can't be deleted/uninstalled without root.
4. Rooting lets you install custom ROMs. A ROM is like installing a different version of the Android OS with some changes/tweeks to it, like changing the layout and design of the user interface (UI), or so that you can get a newer version of Android on your phone before the manufacturer releases an update (if they ever even do!).
These are the main things accomplished by rooting, and
it is important to note that if you attempt to root your phone and don't do it properly, you could "Brick" your phone, which means it becomes inoperable.
Do a forum search for Root and you should find a lot of info. In fact there is a sub-forum for "all things root".
All Things Root - Android Forums