to "root" your phone essentially means to give yourself superuser access on it. This is like being the highest level of administrator for the device. This allows you to change the interface of your phone. Once your phone is "rooted" you can install custom ROMs (they look alot like themes, just google the name and you'll see plenty of examples). You can also download "rooted only" applications for use on the device.
A "Root" in general is the most basic directory in a system. For instance the Root of your computer most likely is the "C:" drive. Their also happens to be a "Root" of your smartphone, which is what is manipulated in order to achieve a "rooted" phone and its benefits.