I'd have to say you're in a bit over your head with your marks on the line. Although it's a great project idea for sure, with your current skill set and understanding of the game in question you will either need some very directed guidance (which perhaps your teacher is providing?) or an alternative project idea.
A few of the users in this thread didn't investigate the game in question, so that's why they have misdirected you to c/c++/java with mention of tcp/udp etc. Meanwhile others suggestions of php/javascript were more on point for the project in question.
Let's say for your project, you keep things very very simple-- no user signups/logins, only a single player with no save state. In that case you could do it all in javascript in the web-browser with no server. You could even make it very easy for yourself and make it in unity/gamemaker or similar game engine, and export to the web-gl.
If you want to go the server route (which is what the game you linked to does) allowing for the possibility of multiplayer, then you need to have some basic understanding of what it even means to have a server in the context of a website. From the sounds of it, you understand basic HTML-- in that case you probably think of the server as just providing static resources like the images and html files. A server can be much more than that as others have suggested, using php on a php enabled server will let you upload php files which are basically programs that are executed on the server computer when someone visits that php page. That program could run some logic and display nothing, or the logic in that program can display some html for the visitor to see as a web-page, among other things.
So just to summarise, when you visit a website, typically there are two different kinds of programs that are running: the 'front end' (ie, javascript program that runs in your web-browser) and the 'back end' (ie, the server-side program that runs on the servers computer, eg in php).
I have a feeling that what I described above probably has gone over your head, which if so I re-iterate my recommendation to search for an alternative project when your marks are on the line ^^
If you feel confident in yourself, proceed at will ^^
If you want to proceed with your initial project and you want an alternative to php, I'd suggest looking at tools such as heroku and implementing the server in a 'RESTful' way. Then you can focus on the front end in javascript and using one of the many front-end libraries available to make your game at least look cool.
If you want to look at a simpler, but related project idea, you could try looking at 'Twine', which would basically let you set up a visual novel/choose your own adventure style game. Less interactive than a true online game, but definitely much more achievable.