Android is built upon the Linux operating system, where there is a concept of an administrator user aka "root" that has full permissions to the system. When you "root" a phone, you are enabling access to the root user so you can call on that user to access protected functionality.
As a regular user, you can request "superuser" access, which is another way of saying that you want root privileges. In a traditional Linux OS, if you log in as root, you can actually create multiple superusers, like a team of administrators, with either complete power, or some subset of power. But for the purpose of Android phones, the only superuser is root, and root always has full access to the system.
To request superuser access, you would use a shell command called "su." In Android, the su command works in conjunction with an app called Superuser to grant the Android application(s) root access. The Superuser app is included in the rooted ROM; it's something you automatically get when you root your phone.
Android applications with root access can do a lot more. If you ever see an app on the market that says "for rooted devices only," it means that its functionality requires root access. A normal unrooted phone would not be able to perform that functionality.