A router is that small box that sits on your desk and is connected to your computer. The purpose of the router is take the information from the modem and deliver it to your computer.
A router is able to take this information from networks and deliver them to several computers. It allows more people to use the internet at the same time using just the one internet connection.
It means you can be working in the office while the kids can be on their computer in their room, both using the internet at the same time.
Having a router is not compulsory but if you run more than one computer then it pays to have one. Otherwise you need to buy an internet connection for each computer.