what is the difference between the admin account and the normal accounts.

Normal users pretty much are limited to their home directory located at /home/$USER or ~/. They are generally free to view files located on the filesystem however.
Admin accounts have more permissions and likely can modify system files. This doesn't mean a admin has access to everything, they may only have access to files located in /etc/ for example. They may have access to sudo which is a selective way of permitting high level actions.
Because of this, what defines a "admin" can be blurred.
In most single person systems (home computer) admin can be equated with root access.

Root is the highest level. root can do anything it wants, whenever it wants. So be careful in root.

Here is something specific to Ubuntu and maybe it will be easier to understand how it works. Non Ubuntu based distributions do not have the root account locked.

https://help.ubuntu.com/community/RootSudo

Oops... someone already posted this link in your previous thread. So the next question is does this make sense.

http://en.wikipedia.org/wiki/Superuser

re wkipedia,
no not all. It is something like forcing people to adopt permisions while making everything ok for me, hence profiles. Back to administator?
Fred.

We have to assume admin is root for this question.

Rights/permissions are the difference. In fact there is no such thing as a normal user. All user accounts below root have less rights/permissions. Generally best practice says to always use systems with the least amount or rights/permissions needed to get the job done.