A device that connects two or more networks and is capable of forwarding data packets to a destination based on information contained within a packet and internal routing tables.
Routers are devices that connect networks and make decisions on where to send incoming data packets, based on the address information contained in the packet and internal routing information.
The most common form are SoHo routers, which connect home or small office networks to the internet and provide routing for all devices on the home/office network. More advanced routers connect large office networks, and at the high end of the scale routers connect large segments of telecommunication networks all over the world at high speed.
Routers are not limited to TCP/IP networks, they can be found in any kind of network.