I see several options to achieve part of what you are asking:
Besically you expose the relevant parts of your apps (web and desktop) via some sort of API and use that for the communication - for example DB and/or WCF (SOAP/REST/whatever).
BEWARE:
If you are after some sort of direct/interactive communication between both apps on the same computer then this won't be possible - at least not without some security related issues.
The browser executes your web app in a sandbox which doesn't allow for direct communication between the web app and the desktop... there are technologies which can help circumvent that (ActiveX, signed Java applets, some Flash technology called AIR etc.)... or you could host some webbrowser/control in your desktop app which in turn runs the web app... BUT basically I would recommend against using any such measures...
IF you really want to go this route then please describe your environment and goal with much more detail.