A privileged user is one who can modify system files, system settings, install software system wide and modify ALL users data.
IIRC, Apple's security model makes a priv'ed user a role that certain users are authorized to take on. When updates are available, OSX notifies you and when you try to update the computer, you must supply your password; this now allows the update program to operate in a priv'ed context and install the updates. Users who do no have this role cannot update the computer.
The privileged account on Unix machines (OSX is a Unix variant) is termed "root"; however Apple has chosen not to allow the "root" user to log in directly. The privileges must be gained from another user account. This is far superior than "another" OS's security model which allows the administrator to log in directly and even surf the web with full privileges invoked because
1) there is no default admin account which hackers know the name of (Windows has an Administrator account name "Administrator" by default unless renamed.
2) Heightened privs are granted ONLY for the actions for which they are needed. User can't "surf the web" with full privs. These heightened privs are available ONLY TO THE PROCESS that needs them.
OSX is good because it has a slick windowing interface built on a rock solid Unix. Apple has chosen to hide 99.99% of the "unixy" stuff underneath and provide a GUI to do everything; The "Terminal" is your window to the underlying OS and rich toolset. You can manage files, edit data, launch applications all from the Terminal window.
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What saved files can be destroyed? Any that are on the computer?
No, just the ones you own. (pictures, MP3's) From the terminal window you can delete files with a command I will not mention here. You also can rename them, copy them, move them between folder; all the stuff you do with the GUI, but it is done with text commands.
Unix is unforgiving; It respects user boundaries between different users, but you are free to blow off your foot with your own files; it will not stop you.
For starters open a Terminal and type
ls
This will list your files. (if you are skeptical or fearful type "man ls" and that will display the man (aka help) page for the ls command.
ls -l (ls minus l) will give you a long list of files (dates included) and so on
Here is a VERY short list of what is available.
http://www.ss64.com/osx/