When you open a socket, you need to address a port of the destination system and open a conjugate listening port on your own system to receive any response. You have to send the destination system your listening port.
Having more than one system using the same modem
When you start a web browser, and go to www.google.com:80, your browser obtains/searches for a free non-system conjugate port from the system for listening. Let us say, the conjugate port is 10000. The listener port is for receiving the http stream back from google.
Then your kid sitting next to you incidentally also browses www.google.com:80 and his/her google session of the play station or xbox-whatever also incidentally is assigned conjugate port 10000.
Both of you are sitting behind a cable modem, and behind the cable modem is your wireless router. And both of your systems are behind the wireless router - All sitting in that sequence, network topology-wise.
To prevent port address collision on the router/modem
Let us say that your cable company DHCP assigns your modem ip4 adress 72.72.72.72. But your wireless router DHCP assigns 192.168.0.10 to your system and to 192.168.0.11 to your kid's system.
When the frame carrying the information of your listener ports passes thro your NAT router, it would translate either one or both listening- ports. Let's say port 15000 for your page and port 16000 for your kid's page.
Your wireless router then sends your requests to google server as coming from 72.72.72.72:15000 and 72.72.72.72:16000.
The google server then responds individually to 72.72.72.72:15000 and 72.72.72.72:16000 and when you wireless router encounters the response, it reaches into the mapping that it has stored and translates 72.72.72.72:15000 to 192.168.0.10:10000 to reach your system but translates 72.72.72.72:16000 to 192.168.0.11:10000 to reach your kid's system.
Running web/game/ftp/etc servers
But what if you have a web server or an ftp server running on your system. What if you have two systems and both have a web server and both web servers are listening on port 80?
Let us say the local ip addresses registered/assigned with your wireless router of your first web server system is 192.168.0.30 and your second web server system is 192.168.0.40.
The wireless router would have a configuration web page usually by default 192.168.0.1:80, unless you changed it. There would be a tab to on the page where you could define/reserve application port mappings.
You could register with your wireless router to reserve the mapping
192.168.0.30:80 => outgoing port 8080
192.168.0.40:80 => outgoing port 8088
So that you have to phone your friends your web/game servers are addressable through
72.72.72.72:8080 and 72.72.72.72:8088 respectively,
where the wireless router would preclude its port 8080 and 8088 from its own dynamic NAT usage.
Of course, 72.72.72.72 is as good as only before your ISP DHCP decide to renew the ip4 address of your modem to say, 72.72.90.200. After which you would have to phone/email your friends and say
Hey, the servers' addresses have changed to 72.72.90.200:8080 and 72.72.90.200:8088 respectively. Or you could subscribe to dynamic dns (ddns) service to use a named domain where the ddns service will need you to install a simple heartbeat utility on your system to help them monitor the address variation. DDNS translation is a separate issue/strategy.
NAT modems
Newer ISP contracts supply you with a modem that has NAT. If so, you have to switch off either the one on your modem or the one on your wireless router. You should not use both - what's the point in translating twice because NAT is simply to prevent address collision. When you switch off NAT from your wireless router, it can operate as a hub switch and not a router anymore so that you could connect it to the modem using one of its LAN socket instead of thro its WAN socket.
