There are two ways to remove a program window from the screen without closing it. You can minimize the window or hide it. Minimized windows place its buttons into the taskbar. Hidden windows are invisible. Many applications use system tray icons (right and left corner of the screen) for opening of hidden windows. But you cannot put an icon of a standard application there.
If you prefer to see your hidden program icon in the system tray you can use ArtTray. After starting this program all other programs will hide into the system tray instead of minimizing.