In a typical home networking setup, internet access is through
a DSL or Cable modem. That modem may be connected to a router, which is
then connected to the networked computers by Ethernet or WiFi. The router
is the device that the Internet sees; it holds the public IP address. The computer behind
the router, on the other hand, is invisible to the Internet as it holds a local IP address.
Port forwarding is necessary in the router because computers will send information directed to
the public IP address and the router needs to know where to send that information.