I am assuming that you already have a connection via Terminal to the school server.
You need to download the .dmg first. Goto the directory in which you want to download the dmg file (command cd \PATH\TO\DIR
)
Then enter this into the terminal:
curl http://mozilla.mirrors.tds.net/pub/mozilla.org/firefox/releases/latest/mac/en-US/Firefox%2025.0.1.dmg -o firefox25.dmg
This is the latest Firefox Release for OS X.
Then mount the dmg:
hdiutil attach firefox25.dmg
Goto the mounted dmg (probably under Volumes):
cd /Volumes/Firefox
There you can find the Firefox.app
which you can now copy to your Applications folder:
cp Firefox.app /Applications/Firefox.app
To unmount the dmg:
hdiutil detach /Volumes/Firefox