Question: probably had these questions a thousand times already.

For a school project I want to make a HTML5 game where you can challenge someone and play against. Now I'm pretty new to game development. I don't now exactly where to start off. There is so much information/technologies on the net that I don't know which to use. I prefer to do this in a known environment (.NET)

I found these: 

* <http://kaazing.com/>
* <http://xsockets.net/>

I also checked out Node.js, socket.io, HTML5 canvas, etc 

It's all a bit of overwhelming for me. :(

Answer: 
Since you're working in a .NET environment, take a look at SignalR, <http://signalr.net>. It's a very nice API around websockets (with fallbacks to other methods for older servers and browsers) that lets you do client-to-server and server-to-client communication.

Code on the client can invoke a Javascript function that will in turn invoke a method on the server. That server method could then send a message down to one or all of the connected clients. Alnitak's answer is correct; your communication would be from client to server to client, not directly client to client.
You will need a central server to relay messages between the two players.

WebSockets (and anything else you might find that plugs into a browser) are unsuitable for direct peer-to-peer communications.