Cannot open Terminal on mac ("Not allowed to use" error message)

Going by the appearance of the dialog window, it looks like the school you got this Mac from was using JAMF Software's Casper Suite to manage their Macs. One of Casper's features is being able to prevent specified applications from launching, with the additional option of providing a message to the end-user. As you're receiving this message, it appears that the Casper agent software is still installed on the Mac in question. There's an uninstall command to remove the Casper agent, but that would need to be run in Terminal. The article at https://jamfnation.jamfsoftware.com/article.html?id=153 by the authors of Casper explains how this is done.

At this point, I think the best bet is to back up your applications and data, wipe the drive of the Mac, and reinstall from scratch. It's the best way to make sure everything which the school had installed is now gone.

Your best bet would be to do a clean install, presuming you have copies / licenses for any software you use on there. Backup any files you want to keep, and follow one of the many guides out there. For instance:

http://mashable.com/2015/10/01/clean-install-os-x-el-capitan/

This could well take less time than figuring out what the IT guys did, and make your Mac faster in the long run.

It looks as if the Terminal app was removed and replaced with an application that displays that dialog, so changing permissions won't help. Doing a clean install as BaronVonKaneHoffen suggested would work, but there are another options.

Option #1 - Restore from another Mac

If you own or have access to another Mac, you can copy /Applications/Terminal.app to /Applications/Terminal.app on this mac.

Option #2 - Restore from the Recovery Partition

(Almost?) Every Mac has had a recovery partition since Mac OS X Lion. If you turn off your computer and turn it back on while holding Command + R as described in https://support.apple.com/en-us/HT201314 you will enter the Recovery Partition. From there, you can select Utilities->Terminal to enter the Terminal. From there, you can type in (untested, something like this)

cd /Applications/Utilities
rm -rf /Volumes/Macintosh\ HD/Applications/Utilities/Terminal.app
cp -r Terminal.app /Volumes/Macintosh\ HD/Applications/Utilities/Terminal.app

Option #3 - Restore from an Apple Software Update

This is the easiest way as it can be done without rebooting or transferring files. Simply download a Terminal replacement such as iTerm (Heck, you might as well just use that instead of Terminal) and run

cd /tmp

rm -rf TERMINALRESTORE
mkdir TERMINALRESTORE
cd TERMINALRESTORE

curl -ORL http://swcdn.apple.com/content/downloads/21/09/031-20634/8d84o1ky5gn2agnf5kiz9eed134n7y3q4c/RecoveryHDUpdate.pkg

xar -xf RecoveryHDUpdate.pkg

hdiutil attach -nobrowse RecoveryHDMeta.dmg
hdiutil attach -nobrowse "/Volumes/Recovery HD Update/BaseSystem.dmg"

sudo rm -rf /Applications/Terminal.app # Erase the old Terminal app. Password required.
sudo cp -Rp "/Volumes/OS X Base System/Applications/Utilities/Terminal.app" /Applications/Utilities/Terminal.app # Copy over the new Terminal app

hdiutil detach "/Volumes/OS X Base System"
hdiutil detach "/Volumes/Recovery HD Update"

cd ~
rm -rf /tmp/TERMINALRESTORE

Additionally, you can simply run curl https://gist.githubusercontent.com/lights0123/2a03d88f99fbbdfc8a35/raw/8d961db047d387319e07e56c228847394ecc1b57/Terminal.command | sh to avoid copying/pasting the above.

It's possible that that is actually a custom dialog box that has been redirected from terminal.app

If you can install apps, look for a terminal alternative program.