Internet Routing devices are generally called as Gateways.Nowadays,the term gateway refers to a device which performs application-layer protocol translation between the devices.
Interior Gateways:
Interior gateways refer to the devices which perform these protocol functions between the machines or networks under the same administrative control or the authority.These are known as autonomous systems.
Exterior Gateways:
Exterior gateways perform the protocol functions between the independent networks.
Interior Routers:
Routers within the internet are organized hierarchically.Routers are used for the information exchange within the autonomous systems and these are called Interior Routers.These routers use interior gateways.The Routing Information Protocol(RIP) is an example of an Interior Gateway Protocol.
Exterior Routers:
Routers that move information between autonomous systems are called as Exterior Routers.These routers use exterior gateways.The Border Gateway Protocol(BGP) is an example of an exterior gateway protocol.
These are called Internet Routing.