I found an app on the internet inside of which, when you click A link it will redirect to A page without refreshing the whole place and changing the url address bar, I know
You can use HTML5 History API to to change the URL without refreshing the page, check out this tutorial from Mozilla https://developer.mozilla.org/en/DOM/Manipulating_the_browser_history
Also check out https://github.com/browserstate/History.js for cross-browser solution
The jQuery PJAX plugin https://github.com/defunkt/jquery-pjax will help you do what you have mentioned, it will help to update the content and change the URL without refresh.
Here is an example: http://pjax.heroku.com/
Source code for this example is available here: https://github.com/defunkt/jquery-pjax/tree/heroku