HTML5 Web Notifications API gives you ability to display desktop notifications that your application has generated.
What you are trying to achieve is a different thing and web notification is just a part of your scenario.
Depending upon how you are managing your application, for chat and messaging purpose as humbolight mentioned, you should look into node.js. it will provide you the necessary back-end to manage sending and receiving messages between users.
To notify a user that (s)he has received a message, you can opt for ajax polling on client side.
Simply create a javascript that pings the server every x seconds and checks if there is any notification or new message available for this user.
If response is successful, then you can use HTML5 notification API to show a message to user that (s)he has a new message.
The main problem with long polling is server load, and bandwidth usage even when there are no messages, and if number of users are in thousands then you can expect your server always busy responding to poll calls.
An alternate is to use Server Sent Events API, where you send a request to server and then server PUSHES the notifications/messages to the client as soon as they are available.
This reduces the unnecessary client->server polling and seems much better option in your case.
To get started you can check a good tutorial at
HTML5Rocks