Hi,
The advantages of rooting your phone are many.
The term root/root'ing is based on the fact that you gain root access to the phone.
In Linux (Android is based on Linux) the root account is the administrator account.
Hence with root you're gaining full controll over your phone.
This gives you the permissions to delete pre-installed apps you don't want/need.
(but be carefull, removing a vital part may brick your phone - rendering it useless)
It also enables you to install apps you normally won't have access to (if you're not in a region with access to Market or the paid version of apps).
It also gives you the possibility to put a custom rom on your phone.
This would be possible later on the mini, work in progress
A custom rom is often a rom based on newer Android versions and have themes/skin and icon packs that the creator of the custom rom likes or think others would like.
Also it has pre-installed and partly configured apps to make the phone "complete" in the eyes of the rom creator (or as a base to build on for others, and in some cases you'll se stripped rom's that contains a bare minimum so those who want a "clean" phone are happy)
Rooting is not for Android phones.
Windows Mobile devices, other PDA's and the Apple iPhone, iPod and iPad also get rooted for the same reasons (and the latter also to drop the iTunes store and be able to install apps not signed by apple)
Hope this helps