So what is root exactly?
In the unix/linux world root is the super user that can do anything on the system. In the windows world the user is called Administrator. See here for a more detailed explanation.
Why should I need/want it?
This is a more difficult question than many people think. In the android world every app runs in its own sandbox, meaning every app can access it's own data and call the system to provide access to system resources but it can't access data from another app. This provides a very good protection against malware etc.....BUT it has disadvantages.
The first disadvantage that comes to mind is the possibility to backup the settings and private data of an app.
The second disadvantage is that for app A to be able to use the data from app B the app B must publish a programmers interface to it's data so the app A can access the data.
Both disadvantages can be overcome with root access.
root access can also overcome some limitations that are build in by the company (in this case Archos) that provided the software that runs on your device.
Read below for examples of what can be done.
Is it dangerous to have root?
The answer to that question is yes if you don't know what you are doing.
If you use root to execute command think twice before executing it. If you are following instructions always try to follow instructions to the letter. If you are not sure about something ask someone who knows about the subject.
If you use apps that require root always read the instructions and if you're not sure try to read about the program in question.
PS: Even the competent root user can make mistakes. I've nuke a few systems in my early years and will never make that mistake twice.
What is su and SuperUser.apk and how do they work?
The su binary is used by other apps to get root access. The SuperUser.apk is a front end to give the user some control over what app can have root.
Whenever the su binary is used by an app it will call upon the SuperUser.apk to pop up a question to Allow/Deny access to that app.