Part of the confusion you're running into is the terminology. In OS X, there are 3 main levels of account privilege you'll run into:
root (aka the superuser, System Administrator, or System): this is a specific account (not just a type of account), and is generally all-powerful. For example, root has full access to all files on the system, not matter what their permissions are. It is generally a bad idea to log in as root, so by default the account is disabled (and mostly hidden). This roughly corresponds to what you're probably calling "Administrator".
Administrator: this is the type of account you have; it's allowed to make system-wide changes, but usually has to do something special (i.e. click a padlock icon and authenticate) to enable that access. For instance, if you try to copy a file you don't have read access to, the Finder will require you to authenticate before copying the file. The
sudo command is another example of this -- it allows administrators to promote themselves temporarily to root.
Standard user: no special privileges, not allowed to mess with (most) system-wide settings or bypass access restrictions.
There's also sort-of a fourth category: managed accounts are standard accounts that have "parental controls" applied to them.