To use a metaphor, a router is like your mailman. It is his responsibility to make sure that all the packages you send reach the post office (your ISP), and that all the packages sent to you are delivered.
To speak literally, a router is a device that forwards data packets along the network and are located at gateways, the places where two or more networks connect.
So in your home, your router forwards data packets to the devices connected to it e.g. your PC, and forwards packets from your PC to the Internet.