how to develop a online chat application for my web site

You can use a server side C# website application with a database the the back end.
And Javascript on the front end.

** The Trick! is to create http requests on the client side but not to return a response from the server immediately. Instead, wait until there is a message to send to the client.

This way you maintain the ability to initiate a call to your clients!

Take notice that this project is not so light, you need to handle session timeouts, and reestablish connections that fail etc.

Good luck.

Cheers,
Edo