Rooting began when the HTC Dream (T-Mobile G1 in the US) was released, and people complained that the developer's version (called Android Dev Phone 1) had more features and more power than what was released to the general public. The first root just included simple tweaks and a few features that weren't available on phones running stock Android. However, as time progressed rooting evolved from simple tweaking of features to giving the Android experience a whole new look and feel.
The process of rooting was originally something that was complex and only those with a knack for technology should do, however the process, much like jailbreaking was simplified to the point where you can obtain root access with just the click of a button. Unlike jailbreaking however, there is no unsigned app store, like Cydia for iOS.
Instead, a program called "Superuser" is the main signal that your device is rooted. What this program does is that it allows certain applications to gain root access to do things like create wireless hotspots, view your device's files, or other functions that can otherwise only be done by Android support staff.
Andy: So? I have a rooted HTC Glacier. I'm running CyanogenMod 6.1.2 and I also have WiFi tethering, a ROM manager, and Overclock.
Isaac: I fucking hate you right now.
Andy: I know, rooting is so awesome, isn't it?
guy: rooting you!!!