Whenever a program wants to "listen" for incoming connections over the internet, it needs to open a "port". The port is a 16 bits integer, so it has a range of 0-65535.
Port forwarding is a simple concept : since your home PCs are behind a router, and the router is the only device directly connected to the internet, it's the one that has to open the port for listening. So for example, if you have a web server (standard port 80) behind a router, only the local computers connected to the router will be able to access it. To enable anyone from the internet to access the server, you would create a port forwarding on your router, so that it redirects (forwards) all incoming internet connections on port 80 to your specified server.
Some games need a port open for listening to incoming connections, especially when they are the kind where you can host a game. You basically need to tell the router to open these ports for listening and forward the traffic to your local PC.
I can be a bit long-winded, but I hope that helped.