Android phones are Linux based, and as such the permissions are very similar. On a Linux system, the user only has access to certain folders, while others are protected.
The term "root" means "root directory", as in the highest folder level on the system. If you were to look at the location bar in Astro or whatever, you'll see things like "/sdcard/Dropbox" or "/emmc/media/video". Start going up until you can't go up any more and the location is simply "/", or "root".
Back to Linux terms. All Linux machines have what's called a "root user" which is a fancy term for an admin account. To install software or modify system settings, you need root permission. If you open up a computer using Ubuntu or Arch or whatever and login as root, you have full access to the machine. Login as a user and you don't.
When we talk about "rooting" the phone, we mean giving ourselves access to modify that root directory and, thus, every single file and folder on the phone, something that we can't do currently.