The router is the "connection-sharing" device that connects directly to your cable or DSL modem. Typical routers have several "wired" ports (Ethernet jacks - like phone jacks, but wider) and often act as wireless access points as well.
Some ISPs build router features directly into their cable or DSL modems.
Your router accepts outgoing Internet requests from computers on your private intranet (the network behind the router) and relays them to the "real" Internet (the network your cable or DSL modem is talking to). Once the router does this, it appears to the outside world as if all of your computers are just one very busy computer.
This is a good thing because it prevents outsiders from connecting directly to your PCs, which is much more dangerous from a security perspective.
For personal use, it doesn't matter if you have a "dynamic" IP address (assigned "on the fly" each time your cable or DSL modem connects) or a "static" IP address (permanently assigned). Static IPs are usually only helpful to those who are choosing to run servers on their own computers. And even then, you can work around the need for a static IP address by using a dynamic DNS service. For more information, see the articles should I host my own website at home? and how can I host my own web site at home?
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