Here is my go at an explanation, let me know if you can't understand it. Note: Some of this is not 100% correct technically speaking but describes the principals.

What is a port

Information from network applications flows across the network from one computer to another. There may be many different applications on the same computer using the network.

To be able to distinguish between information destined for each application a port is used. A port is a number that is added to the information to tell it what application will use it.

Its a bit like sending mail to a block of flats. The mail is sent to the block of flats (the computer) and is then put in a mail box with the specific flat's house number on it (the port). The person at each flat (application) can then get information from the mail box.

What is port forwarding

Some routers protect the computer by not allowing information through that the router does not recognize. It does this for all information regardless if it is harmful or not.

If you want a game server to work you need to tell the router the game information is actually safe and that you want it. It will then allow the information through. This is called port forwarding.

To set up port forwarding, you configure the router to send all game information (information addressed to the game ports) to your computer.