A router is a network device that works at Layer 3 and routes packets by looking at the IP destination address of incoming packets. It compares this with entries in its routing table and decides which network (router) is the best one to forward the packet on to.
Routing tables do not contain lists of all computers, but only the routes to get to particular networks.
Routing tables used by routers will have entries for directly connected networks. Information about other network can be added manually (static routing) or automatically using a routing protocol such as OSPF or IS-IS.
