Hide iTunes when minimized: there are two related settings you can configure in iTunes Preferences to make iTunes hide itself once minimized (no visible button in the Windows taskbar), and another option that allows you to completely hide the iTunes icon from the system tray (notification area). This tutorial focuses exclusively on Windows, since Mac users have a native command to achieve the same thing, described below (and also because there is no system tray on Mac OS X, making the second settings irrelevant to Mac users).
On a Mac, you can hide iTunes simply by going to iTunes (menu) > Hide. This is not an iTunes-specific command, and appears in the Application menu of all apps running on Mac OS X; alternatively, you can use the keyboard shortcut equivalent of Command+H.
Hide iTunes when you minimize it in Windows
From iTunes' main window, go the Edit menu, and choose Preferences; now click on the Advanced tab. Locate the last two checkboxes of this tab; if needed, check the "Show iTunes icon in system tray" to enable the checkbox below it; finally, check the "Minimize iTunes window to system tray" checkbox and click OK to apply your new settings. From now on, iTunes will be hidden from the Windows taskbar when you minimize it.
Since Windows Vista and Windows 7, the "system tray" is now called "notification area" — both are, and refer to, exactly the same thing, namely the area of the taskbar next to the clock; the new name is just less technical-sounding, and Microsoft found it more user-friendly.
To restore your iTunes window, simply double-click on the iTunes icon (this is why the "Show iTunes icon in system tray" setting has to be enabled to make iTunes hidden when minimized).
Alternatively, you can right-click on the iTunes icon and choose "Show iTunes" from the context menu that appears, revealing also additional functionality you might often need to access, without having to restore the iTunes window: change the current loop / repeat mode, shuffle the order in which iTunes plays your music (using your current shuffle settings).
From the iTunes icon context menu, you can also of course Play or Pause your music, switch to the Previous or Next track in the current playlist, or Mute iTunes: this setting temporarily affects iTunes' playback volume, not Windows' current sound volume level.
Finally, you can Exit iTunes from the last item in its icon's context menu.
Hide the iTunes icon from the system tray (notification area)
If you are not interested in using the feature described above to hide iTunes' window when minimized, you can uncheck the "Show iTunes icon in system tray" checkbox - this will hide the iTunes icon that otherwise would be visible in the system tray / notification area, next to the system clock. And in this case, iTunes will not hide itself from the taskbar when you minimize it.