They transmit their internal tables which includes IP addresses of all possible destinations and IP addresses about the next router. Thus, changes made in a sub-network get transmitted to the entire network so the routers can update their tables. In link-state routing, each router shares its knowledge about its neighbors to all routers. It usually does this only at startup and when there are changes. A gateway is a protocol converter.
Routers only transmit across networks using similar protocols.
Gateways may be hardware boxes, or software within a router. For example, to connect your internal network to the Internet.