You are looking for the NotifyIcon class: http://msdn.microsoft.com/en-gb/library/system.windows.forms.notifyicon.aspx
and here is a tutorial: http://alanbondo.wordpress.com/2008/06/22/creating-a-system-tray-app-with-c/
That said just adding your application to the system tray is a really bad UI idea. It used to be very popular during the windows 95/98 days and Microsoft helpfully provided a working code example that everyone copied and pasted. Soon we had fish and eyes swimming in our system trays. And the tray got fuller and fuller and hence completely useless.
If your application is doing something that does not need user attention, make it a service, or just let the user minimize the dialog to the taskbar and show a progress bar on the dialog. That way hovering over the taskbar icon will allow the user to check progress.
Use the system try to show helpful notifications. Add an icon only if the icon will convey a system level activity that could effect the users current work (like network connectivity). Otherwise, don't clutter my system tray. I don't want to take my eyes off the work that needs my attention (you said yours doesn't) to look at a bouncing bunny in my system tray.