A router (computing) is nothing but a networking device that is responsible for the forwarding of data packets from one computer to another. To put it simply, these devices perform the function of traffic directing across the web. Typically, a data packet is forwarded from one computer to another through a network till it reaches its final destination.
A router (computing) is connected to a couple or more data lines that are part of different networks. This is what differentiates them from network switches which connect data connections to one single network. Once a data packet reaches a router, it access the information embedded in the packet for determining the ultimate destination. Now, using this information it channelizes the packet in the right direction.
There are many types of this device made available. For instance, you have simpler versions that are used in small offices and homes and you also have enterprise routers for large scale usage. Cable or DSL versions are more suited for home use where their task is to build connections to the internet through an internet service provider. Larger capacities ones are more sophisticated and are capable of connecting large business houses to their internet service providers.