Think of your router as being a huge electric fence or wall, with a few doors or openings. This is called Port Forwarding If you have a router at home or your office and you want to access it remotely, you will need to forward ports in order for outside traffic to get into your network.
So, in order to run a mail server, game server, access your computer remotely, etc you will need to open an extra door or two in your router in order for the outside traffic to get inside. Your router comes preconfigured with a few of those doors (or ports) open to let you access the internet, but the others are closed tight. This electric fence, or wall serves as your barrier and security blanket from the scary outside Internet world.