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Before we talk about the wireless portion of the wireless router, let's first cover what a router does.
A router is a piece of networking equipment which is used to allow numerous computers to communicate together. Obviously, most computers have only on Ethernet jack for wired connections and have only one wireless antenna for receiving an sending wireless signals. Because of this, a network of computers can't communicate simply by speaking directly to each other. They need something in between to manage the traffic, and that is the job of the router.
Routers also serve as an access point to the Internet, allowing multiple computers to share one Internet connection. They do this by intercepting information sent from computers to the router and then forward that information to the Internet. When that information returns, the router intercepts it again and sends it to the computer that sent it. Without this it would be impossible for multiple computers to use the same Internet connection because there would be no way to know which computer sent out what data.