How does Stack Overflow show the answer added/edited message without a page reload?
To add to Xavi's answer, if you want to get started with web sockets without having to understand all the internals, you might try out Pusher, a library for multiple platforms (including PHP) that makes push notifications on the web very straightforward.
I do not work for Pusher; it's just a product I've found very useful in the past. I've always used the free version for small personal projects, though I would probably pay if I ever used it on a larger application.