Port forwarding is basically a way to establish a direct connection that's uninterrupted between a computer and your network, and the outside world, or a service on the Internet. Why would you need that? Well, most of us use these guys to connect to the Internet, and these are routers. And they basically get in the way between your computer and the service on the Internet.
Now, this is usually a good thing because it acts as a Firewall, it stands in the way between your computer so that outside sources on the Internet, all the bad guys, can't see your computer directly. All they can see is the router, and the router kind of filters through all the communication that comes from the Internet to your computer.
But sometimes that's a bad thing because some programs need a direct connection to the Internet. Stuff like, video camera monitoring, certain games, even virtual private networks, they all need a direct connection that's uninterrupted by a router or firewall. And that's what port forwarding is.
Port forwarding is basically a way to go and tell your router, listen when programs come in through that specific port on the Internet, always forward them to this computer instead of that computer. So it's a way to basically assign a service to a specific computer on your network. You do it through the router set up page, and once it's done, it's set for good, and your computer will always know that this service and this port belongs to it.