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);
            }
        );
    }
);