The term "Internet Channel" for Wii actually refers to WiiConnect24, a background service rather than a visible channel on the Wii Menu. It allowed the console to maintain a low-power internet connection even when turned off, enabling several features:
Essentially, WiiConnect24 acted as a constant, low-power link to the internet for a more dynamic gaming experience.
WiiConnect24 was officially shut down on June 20, 2013. The service no longer functions; you cannot download updates, purchase Virtual Console titles, or use any online features that relied on it. A Wii console today is limited to local/offline gameplay only.