Most multi-player games work by having the client application connect to a games server.
The client application can be just the browser connecting to web server serving up html/javascript pages, or browser running a Java App, up to a fully fledged application that has to be downloaded and installed.
Teaching you how to do this is a bit beyond the scope of these forums however.

Also if you hit Google, you'd probably find a bunch of free tutorials and networking libraries out there for the same.

dijk: Thank you! I didn't expect to learn how to do it at the forums--but knowing it's an ok solution to have 2 browsers run a java app & connect to a games server will make searching a lot easier.

Is a games server different from the one already serving up my html pages?

sabrok: will do!

There are a few open source game servers out there.

You might investigate Project DarkStar [projectdarkstar.com] sponsored by Sun to start with and go from there.

It's shockingly written entirely in Java, what else!

bill--it's looking really helpful, thanks! They link to lots of user-made tutorials, which is a big bonus for me.

And I love your username.

I think I'm going to start with making a basic two-person chat program and go from there.

Best wishes! Do take all the time necessary to understand the security issues involved. You do not want your game server/app a doorway to providing malware to you visitors.