Router
Routers (in current terminology) are the packet switches that form the lowest layer of internetwork networking infrastructure (along with the physical networks that connect them).
Originally they were called gateways, but this name was eventually superseded, because it was used for a variety of kinds of devices, potentially causing confusion.
Routers operate at the internetworking layer, the protocol which offers direct datagram carriage across the entire network, providing an un-reliable service which makes no guarantees that packets will not be damaged, delayed, duplicated or re-ordered.
In the TCP/IP protocol suite (the protocol family of the Internet), that protocol is the Internet Protocol.
Routers usually implement a variety of other protocols as well, for operation and maintenance purposes, such as the somewhat confusing named routing protocols, which are used for path selection.