What ways are out there to display a desktop notification from a web app?

Solution 1:

Below is a working example of desktop notifications for Chrome, Firefox, Opera and Safari, copied from Chrome desktop notification example.
Try it live on JSBin.

// request permission on page load
document.addEventListener('DOMContentLoaded', function () {
  if (Notification.permission !== "granted")
    Notification.requestPermission();
});

function notifyMe() {
  if (!Notification) {
    alert('Desktop notifications not available in your browser. Try Chromium.'); 
    return;
  }

  if (Notification.permission !== "granted")
    Notification.requestPermission();
  else {
    var notification = new Notification('Notification title', {
      icon: 'http://cdn.sstatic.net/stackexchange/img/logos/so/so-icon.png',
      body: "Hey there! You've been notified!",
    });

    notification.onclick = function () {
      window.open("https://stackoverflow.com/a/13328397/1269037");      
    };

  }

}

<button onclick="notifyMe()">Notify me!</button>

More information about how this works in my answer to Chrome desktop notification example.

Solution 2:

On Chrome, you can achieve something like this using Desktop notification. See Chrome desktop notification example for example.

If your users have growl installed on their machine you should check out growl developer page