This part is called the system tray on the taskbar. The NotifyIcon
component allows you to show an icon in the system tray. To
have your window "minimize" to the system tray, when it is
minimized you set its ShowInTaskbar property to False and the
NotifyIcon's Visible property to true.
You can create a ContextMenu that opens the form, and then set it to
the NotifyIcon's ContextMenu property. When the user right-clicks
the icon, the menu will pop up.