I don't think you've quite got the router bit right. What you've all described is a hub or switch.
A router has additional intelligence built in to enable it to decide where to send a network packet if the packet is addressed to a network other than the local one.
It acts as a sort of 'automatic telephone exchange' for networks. Your computer can do the local calls itself, but if calls to other exchanges are made, the router decides how to handle it. IE it routes the message.
In home use, it isn't really very complicated to do, as if the message isn't for the local network then it's for the ISP, at the other end of the ADSL link. In commercial use, with lots of different networks, it get a bit more complicated.