Web Push
Web Push API’s allows server-side applications to push messages to browsers.
The browser must first subscribe to the push notifications for the server app, and then it will keep a Service Worker available (details depend with the operating system) to receive and react to the notifications.
For now, the big missing piece for this is iOS, it doesn’t support it. Chrome supports it everywhere else.
Specs
How to use it
The simplest way to start with Push API is to use the OneSignal platform for now.
The main SDK documentation, the one you’ll include on your WebSite, is here.
To understand what OneSignal hides from you, you can read the detailed flow on Push workflow summary. A step-by-step tutorial is available on Your first push notifications web app.