The first account created (when initially setting up MacOS X) is not the 'root' account; but, just an 'administrator' account. It is then from this administrator's account - that other accounts (even designated as 'administrator') can then be created.
To create a 'root' account, the current administrator must launch 'NetInfo Manager' ('/Applications/Utilities/') and manually set up the 'root' account.
Thus, did your father merely set up the initial 'administrator' account; or, did he also set up the 'root' account?
'It gives us the error of "Unsufficent privelegeds" or something.' is insufficient. Try unzipping a file, and this time record exactly what is displayed - then reply with such information.
If your father did set up his administrator's account for 'root' access, have him disable the 'root' account. Then see if you can once again unzip any '.zip' files.
