Yes, as goldilocks suggests you need an ip,
Assuming you are already connected to the internet and can browse etc, you will need to somehow route the inbound traffic 'calls' (like from the students' computers at home calling the classroom server), and could be achieved a few ways.
You might have to have a net admin setup the router to forward an external port to your internal 8081 port. So you can assign say a port like 9001 to listen for those external calls to your server.com:9001 and when the user connects, the router will internally direct 9001 to 8081. You have to set this up in the router and is the more longterm, more difficult, privileged way of setting it up. Talk with your net admin at your location who has privileged access to your wifi router. This might help if you choose this method.
OR: You can use ngrok.
Lookup how to install ngrok on your machine. (Linux for Pi!)
After which setup is easy. Basically it is a process that will need to stay running on the pi. You will type something in terminal to run it like:
./ngrok http 8081
This will display a temporary random http and https address you can visit that will direct right to your pi running that ngrok process. This will last as long as that process stays running. If you restart the pi or that ngrok process, you will get a new address. This is where they make money if you buy a permanent address, but if you leave the pi on all the time, and link the ngrok address from a shared link like a facebook link, you might be ok without having to pay.
I use this ngrok all the time to open a tunnel on my local machine, so I can visit my web-apps without needing a hosted webspace and domain name.
You can port all type of web protocol.
As far as a 'User Login' system,
I've had decent luck with a motion based package called MotionEye that you may or may not find easy to setup. But once you do get it going, you can setup a Watch Only user account so your students can log into that to watch the feed. There might be other methods like .htaccess password protecting your webpages, or someone else might have a nifty way.