It's just root access to your phone-- like having administrator access to your desktop computer. With root access you have permissions to do things you otherwise wouldn't-- like install a tweaked version of the OS.

It's pretty easy to root your phone at this point, and not too likely you'll eff it up. But it is a possibility, so if you do so, do plenty of research and if following a tutorial make sure you read through it first, understand the steps, etc.

Here are a couple resources:

One Click Root Access via Lifehacker

Tutorials on Cyanogen's Wiki

Short answer:

"root" on Linux = "Administrator" account on Windows.

Long answer:

A smart-phone that is based on variation of Linux (Android or WebOS, for example) will use an account that has limited privileges, so that the end-user doesn't go around deleting important system files from their devices. Such accounts are usually locked down very tightly in terms of being able to do things on the device that the carrier restricts (tethering or installing apps onto SD card, for example) even though the phone is perfectly capable of doing that. That is of course done to milk more money out of us end-users, so that the carriers can charge for things like ringtones, background wallpapers, and other similar things that the phone is capable of doing if not for these arbitrary restrictions.

For advanced users this limited account means having a crippled device, that is at the mercy of manufacturer or carrier. The act of "rooting" a phone means gaining access to the "root" account that has administrative privileges, and thus unlocking the true potential of the device. Since carriers don't like it when you bypass their blocks, they make it extremely difficult to gain root access. That is why rooting usually requires exploiting a bug in the phone's original firmware.

In case of T-Mobile G1, it was accomplished by the fact that early OS versions had a flaw where Terminal (an equivalent to Command Prompt in Windows) would intercept anything typed into the phone as actual commands and execute them, allowing installation of restricted software. Google since patched this vulnerability, which is why the first step of rooting new G1s requires a downgrade. In order to root other Android phones similar vulnerability that can be exploited must be discovered first, which is why it takes a while to root them.

Since under shiny Java/Dalvik covers there are Linux workhorse "root" simply means *nix superuser. Which also means you can get into deep trouble really fast :) With the power comes responsibility I guess...

Don't you just love it when someone defines a word while using the word itself in the definition!

ROOT means to unlock your phone and give you 100% unrestricted access to modify anything you desire on the device.