To understand what the purpose of a router is, we also have to understand how the internet functions. It may seem a little complicated, but when you break it down, it really isn't.
The internet is a wide area network of different servers and different networks all over the world. They connect to each other at millions of different points, making a web, hence the name, World Wide Web. To talk to each other, these networks use a specific language called TCP/IP, which stands for Transmission Control Protocol/Internet Protocol. This language is designed specifically for networks, and allows all the different devices in the world to talk to, and understand one another.
The fundamental part of internet traffic is called a packet. It contains a header, payload and other information not relevant to us right now. The header is like the destination zip code. It is the IP address of where the packet is going, which tells the network where to send it. The payload is the actual data being transported, which could be part of an email, a file segment or a request for a new web page, anything really.
Networks, large and small are all controlled by routers. They route traffic, hence their name. They have large databases called "IP Tables" which can hold millions of IP addresses. The router looks at the header of a packet and checks to see if that IP address is connected to it. If not, it looks to see which of its other network connections is the quickest way to that destination. It chooses the next router in the chain and sends it along. This process is repeated all along the chain or "network path" until the packet has reached the destination computer. While this may seem a laborious task, it actually happens lightning fast, and only adds milliseconds to the average journey time.