How can I do push notifications in an HTML5 Web application?

You can do real push notifications with Web apps today in Chrome using Service Workers and PushManager from the W3C Push API.

See the article Push Notifications on the Open Web for a step-by-step how-to and code snippets you can use. Here’s a diagram from that article that explains what the UI around it looks like.

An implementation of the Push API has already landed in Firefox too; it’s targeted for shipping in November 2015 in Firefox 42. And Microsoft has indicated that the Push API is also under consideration for implementation in Edge team as well.

Below is a simple code example, borrowed from MDN.

this.onpush = function(event) {
  console.log(event.data);
}

navigator.serviceWorker.register('serviceworker.js').then(
    function(serviceWorkerRegistration) {
        serviceWorkerRegistration.pushManager.subscribe().then(
            function(pushSubscription) {
                console.log(pushSubscription.subscriptionId);
                console.log(pushSubscription.endpoint);
            }, function(error) {
                console.log(error);
            }
        );
    }
);

It depends on what you want to achieve:

• if you want to display a push notifications to the user while is surfing your website you can use the Web Notifications API, to give the notification a "native" style; you may also use a technology like SSE or WebSockets to push the notification from your server to the client
• if you want off-site push notifications (that are delivered even when the user is not on your website) you should use a combination of Service Workers and Push API to trigger the offline event and use the Notifications API to display the notification (see my answer)