Frames / Iframes
One way to achieve this is to create a frameset or use iframes to show the sub pages. The top (or bottom) frame would be a very tiny one, and containing your music player. The big frame would contain the page that is being browsed. This technique is used e.g. by Google when clicking on an image search result.
This method has its downsides: The user will see your URL in the address bar and not the one of the page being browsed. If the user enters something in the address bar, they will leave your frameset. It is not possible e.g. to write down the URL of the current page.
Ajax
The second, better way is to build a navigation that loads the other pages into the current page through AJAX. See an example implementation here.
This will provide for smooth loading, music will continue to play. If done right, it is even possible to retain a healthy link structure that won't break external references, and have the "back" button work. The tutorial I link to covers both aspects. Just be careful, it's a 3-part tutorial.
It works with JavaScript enabled only, though, but there are solutions that downgrade gracefully (falling back to the "normal" behaviour of switching pages when JavaScript is turned off).