I recently had this question posed to me by an individual who was getting a Top Notch Tablets rom from the site and they noted it was "rooted".
I have seen a lot of posts about "rooting" (aka "jail breaking" in the Apple world...) and "root" access and "super user" that were highly erroneous. People tend to think that rooting is some mystical/magical thing on a device when it is nothing of the sort, but rather a regular function that is part of the Linux underpinnings of android.
So, here was the answer that I gave as way of explanation:
Android is built on Linux, an open-source operating system (like Windows, but the code is freely available for everyone to use and modify as they like).
In linux, there is a user called "root". So called because this user account has access to the "root" of the file system, all files, and all administrative functions. Like a super administrator for the system.
By default, when you are on an android device you are "logged in" as a regular user and don't have full control of the system. This is for security purposes as malicious code and applications that could accidentally be ran by you would only have limited privileges and would be limited in the harm they could cause.
When a device is "rooted" it means you have the ability to "become" the root user on the system whenever you need more privileges to perform a certain action. In this case though, we use a "super user" program/app to regulate this. So anytime an application needs "root" access or "root" privileges, a prompt will pop up and ask you for permission to allow the program to run as root (example pictured above). This forces the user to make a decision and be aware of any instance when a program needs root access to the system. Otherwise, programs will be run normally, with limited-privilege access.
Finally, as long as you are getting most/all of your applications from the Play Store, there is very low likelihood of downloading any kind of malicious program that is going to abuse root access.
The nice thing though, is that programs that do need root access (certain backup programs for example) can be given it, whereas before the device was rooted, they could not. If you like to tinker, you can now also get into the root file system and mess with files. I recommend not doing this though unless you don't mind messing things up or know what you are about.
So I think that is a fairly accurate answer as to what "root" actually is. Hopefully this will dispel any myths around it. Now, how you go about "rooting" a device that isn't rooted by default in the firmware, well... that is a bit trickier and differs from device to device. That is also well outside of the scope of this post.
Cheers and Happy New Year!